conference | James Colliander

IAS Workshop on Symplectic Dynamics 2: Friday

Fri, 16 Mar 2012 18:49:10 +0000

IAS School of Mathematics

Workshop web page

Friday: 2012-03-16

9:00 - 10:00 James Colliander, University of Toronto, “Big frequency cascades in the cubic nonlinear Schroedinger flow on the 2-torus” abstract
10:15 - 11:15 Marcel Guardia, IAS, “Growth of Sobolev norms for the cubic defocusing nonlinear Schroedinger equation in polynomial time” abstract
11:30 - 12:30 Yann Brenier, University of Nice, “Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics” abstract

James Colliander: Big frequency cascades in the cubic nonlinear Schrödinger flow on the 2-torus

(chalk talk)

(joint work with M. Keel, G. Staffilani, H. Takaoka, T. Tao)

I prepared slides but decided to give a chalk talk. The slides are located here: http://uoft.me/nls-cascade. The paper discussed in this talk is located here.

(See also: The thesis of Zaher Hani has advanced along these lines and is surveyed on his slides from the Ilde de Berder Workshop.)

The construction of the frequency civilization is partly conveyed by the following cartoon. Notice that the underachieving child frequency in the cartoon is always sent to the zero frequency. This violates the injectivity requirements used in our construction of the set $\Lambda$.

The next cartoon is meant to convey a traveling wave through the generations in the civilization. This wave is constructed by concatenating heteroclinic orbits in the toy model evolution.

The idea that the orbits could be concatenated reminded my coauthors of this famous commercial:

http://www.youtube.com/watch?v=KXA8g90g7so

Marcel Guardia: Growth of Sobolev norms for the cubic defocusing nonlinear Schrödinger equation in polynomial time

(joint work with Vadim Kaloshin; we have a preprint; slides from the talk; 32 pages)

This talk is strongly related with the previous talk.

$NLS_3^+ (T^2)$. Energy and Mass are conserved. The problem is globally well-psed in time Bourgain 1993.

Transfer of Energy

Fourier series of $u$.
Can we have a transfer of energy to higher and higher modes ass $ t \rightarrow + \infty$?
This is quantified with the growth of Sobolev norms.

We need to move mass toward high frequencies in a careful way to satisfy the mass and energy constraints.

Theoreom (Bourgain 1993): As $t \rightarrow + \infty$, the $H^s$ norm is upper bounded by $\leq t^{2(s-1)+} | u(0) |_{{H^s}}.$

This result has been improved or applied to other Hamiltonian PDEs by various authors.

Question (Bourgain 2000): Are there solutions $u$ such that for $ s>1$ such that $$| u(t)|s \rightarrow \infty $$ as $ t \rightarrow + \infty? $ Moreover, he conjectured that the growth should be subpolynomial in time: $ | u(t)|{H^s} \ll t^\epsilon$.

The second part was partly conjectured because of insights related to Nekoroshev type theorems for NLS.

Kuksin studied the growth of Sobolev norms for NLS for large initial condition. For such data, a change of coordinates recasts the dynamics into

i \dot{w} = - \delta \Delta w + |w|^2 w, ~ \delta \ll 1. $$

Theorem (CKSTT 2010): $\exists $ big frequency cascades in the $NLS_3^+ (T^2)$ flow.

The solutions have small intial mass and energy. They remain small as time involves whereas the s-Sobolev norm grows considerably.

Theorem: (long statement, I’m reading instead of typing.)

The mass is small but the $H^s$ norm is initially large. They can then grow it up to any big threshold over a polynomially related time interval.

Remark: One might view this equation as a perturbation (when the data is small) of the (integrable) linear Schr"odinger. It is well know that the Nekoroshov type results for PDEs often loses the exponential estimates and becomes polynomial. Our result is consistent with this.

Remark: Our result deals with a different regime than the Bourgain subpolynomial conjecture. Our result is rather fast, but it could perhaps slow down over infinite time. Our construction involves a finite number of modes. If we try to build something on an infinite number of modes, the transfer mechanism might slow down.

Comments:

One can tensor this up to obtain similar results on $T^d, d \geq 2$.
We can obtain more detailed information about the distribution of the Sobolev norm of the solution $u$, among its Fourier modes when $t = T$. In particular, the high Sobolev norm is carried by two high achievers at the last stage. The high Sobolev norm is essentially localized in two modes.

Main Ideas in the Proof:

$I$-team introduced a finite-d toy model.
This toy model approximates well certain solutions of NLS
Our contribution is the analysis of the toy model. Using dynamical system tools, and a careful choice of the initial conditions, we find a faster motion.
The solutions of NLS can be proven to approximate well the solutions for the toy model for long time.

Reduction to the toy model.

$FNLS$
$RFNLS$
Construct $\Lambda$.
Toy Model ODE

For $N$ big enough, the set $\Lambda$ can be chosen to have the “wide diaspora property.” This is partly why we don’t have an infinite cascade. The construction only involves a finite number of modes. We want to quantify everything in terms of the number $N$ of generations. At the end we have $N \thicksim \log K$. We have to quantify everything.

Toy Model Theorem: There exists an orbit in the toy model which moves from the first generation to the last. Their statement includes quantifications! They compute the time of this transfer process.

To make things happen quickly, they want to make the transfers as fast as possible. This development uses a different orbit construction than the one performed by CKSTT.

Dynamics of the Toy Model:

ODE explicitly written out.
Each 4-d plane is invaraint.
Dynamics in each 4-d plane is given by a simple Hamiltonian involving nearest neighbor interactions.

Nice picture of invariant planes intersecting to form something like a polyhedra with a curve following along nearby invariant lines. “Of course, we are not in the plance but we are nearby it.” Of course to do this, we need to understand the dynamics in each of these planes. To obtain these orbits, we use hyperbolicity. But these planes have certain normal positive Lyapunov exponents so one has to be very careful. If we just move away from these planes, we lose control.

Dynamics in $L_j$:

To construct such orbits, we need to understand dynamics in each $L_j$.
Hamiltonian $h_j$ and $M_j(b_j, b_{j+1}) = |b_j|^2 + |b_{j+1}|^2$…..ack slide changed.
Contains two periodic orbits.
Periodic orbits in $L_j$ are hyperbolic.
Stable and unstable invariant manifolds of the periodic orbits coincide.
Call $\gamma_j$ the heteroclinic connection between the two dimensional manifold asymptotic to $T_j$ as $ t \rightarrow - \infty$ and asymptotic to $T_{j+1}$ as $ t \rightarrow \infty$.

Key Problem: The Shadowing

(nice picture)

We put sections transveral to the flow.
We study local maps: dynamics close to the periodic orbits $T_j$. Global maps: study dynamics close to the heteroclinic connections $\gamma_j$.

Local and Global Maps:

Shadowing for global map is basically applying (refined) Gronwall estimates.
Local map is more delicate: periodic orbits are of mixed type. Hyperbolic eigenvalues are resonant.
This resonance complicates the analysis of the local maps.

We need to choose very carefully which orbits we study.

The Model Problem:

After some reductions, we have a Hamiltonian of the form:

$$ H(p,q) = p_1 q_1 + p_2 q_2 + H_4 (p,q) $$ where $H_4$ is a degree 4 homogenous polynomial, the variables “1” correspond to the variable $b_{j-1}$ ….slide changed.

Analysis of map from a section $\Sigma_+$ to $\Sigma_-$.

Dynamics of the linear saddle (Kill the $H_4$ and see what happens.).

Dynamics of the resonant saddle:

System is not well approximated by its linear part due to the resonance.
For typical initial conditions, we have a resonat affect creating logarithmic (in $\delta$ ) corrections to the transfer across hetereoclinic connections.
We need $~N$ transitions.
The number of logarithms becomes exponential in $N$.
We need to stay close to the periodic orbits to control the shadowing
This implies we need to start….slide change

We use the beautiful Shilnikov trick. The worst term that was developing with logarithms is now computed more accurately in terms of some function $g(p_0, q_0)$. This transfers the resonant saddle dynamics into essentially the dynamics of the linear saddle, provided that we carefully choose the domain of the map. This is kind of delicate and needs to be iterated through compositions.

Composing the local and the global maps:

We need to compose the local and global maps.
We define sets $U_j$ in the transversal secions and we show that the dynamics moves one into the other. (This is the “perfect shot”.)
To avoid deviations at each local map, we need to impose a restriction at every step.
“Product-like” step.

Product-like structure sets.

We start with a polydisk.
At each step, we impose a condition on the mode $b_{j-1}$.
Inductively, we rstrict the domain on previous domains involving conditions on previous mode involving the Shilnikov function $g$.
Since the restricitons involve a different mode at each step, the conditions are compatible.

Composing the local and global maps produces the toy model result. The detailed discussion partly explains the time quantification.

Approximating solutions of NLS:

Last step obtain a solution of NLS close to the solution of the toy model.
We modify the set $\Lambda$ from the $I$-tema so that the modes out of $\Lambda$ only gets influenced by few modes in $\Lambda$.
Each $b_j$ is excited only for a short period of time.
A mode out of $\Lambda$ only receives mass from $\Lambda$ during a short time.
This implies that the spreading of mass to modes out of $\Lambda$ is very slow.
We obtain an orbit for NLS that undergoes the growth of Sobolev nroms in polynomial time.

Yann Brenier: Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics

(chalk talk; the slides once linked here are no longer available.)

It’s a good time for all of us to thank the organizers for this meeting. (Applause!)

Related to a question posed by Shnirelman from 1985.

System of interacting particles along the real line with sticky collisions. When the particles hit, they merge and continue with the same momentum. This is an inelastic, sticky collision. This is clearly

dissipative
nonreversible in time

Shnirelman’s Question (1985): Can we modify the action principle to handle these dissipative collisions?

Unfortunately, the paper is hard to find. You can think of the collision in a higher dimensional space and keep track of the energy in the extra variables.

G. Wolansky (2008 ?)

In this talk, I want to provide some ideas that come from ideal fluids. This seems strange because this problem is highly compressible, etc.

This talk is about a proposal for a modified action suggested by ideal fluid mechanics.

Arnold’s geometric interpretation (1966) of Euler equation for incompressible fluids (1755).

Let $D = [0,1]^3$. Let $VPM (D) = [ volume ~ preserving ~ maps ~ of ~ D]$. This may be viewed as a subset of $H = L^2 (D, R^3)$. Geodesics along VPM are (formally) the solutions of the Euler equations.

There is a discrete subset of $VPM (D)$ are the permutation maps $S =P_N (D)$. Partition the unit cube into a collection of $N$ subcubes $Q_i$ each with center of mass $A_i$. You would like to do some kind of discrete fluid mechanics by exchanging these cubes. There is a folklore of approximating geodesics with these kinds of maps. This is used in some works in computational geometry. How to define approximate geodesics along $P_N (D)$?

More generally, let $H$ be a Euclidean (or Hilbert) space. You have a closed bounded subset $S$. Introduce a potential $$\Phi [x] = \frac{d^2}{2} (x,s) = \inf_{s \in S} \frac{|x-s|^2}{2} = \frac{|x|^2}{2} - R(x). $$ Here $R$ is the Legendre transform: $$ R(x) = \sup_{s \in S} (x|s) - \frac{1}{2} |s|^2. $$ Convex, Lipschitz, usually not smooth.

Approximate minimizing geodesics are found by minimizing between two given points $A, B \in H$ by $$ \int_0^1 (\frac{1}{2} |\frac{dx}{dt} (t)|^2 + \frac{1}{2\epsilon} \Phi [x(t)] ) dt $$ satisfying $X(0) = A, X(1) = B$. If $S$ is a smooth manifold this converges to geodesics Rubin-Ungar 1957 (Yann’s birth year!).

These ideas were applied by David Ebin to fluids.

A simpler example than the one appearing in Shnirelman’s question…

Take $H = R^2$. Let $S$ be the St. George cross. He writes the coordinate axes in $R^2$ in red and forecasts that a joke will soon come up…

Whenever $\Phi$ is smooth about $X$, we have $\nabla \Phi (x) = x - \pi_S (x)$ (the closes point to $x$ inside $S$, not necessarily unique). The bad set $N$ where differentiability fails is both meager and has lebesgue measure zero in finite-d case. This has to do with the regularity of Lipschitz functions.

What is the bad set related to the St. George cross? Of course, it is the St. Andrew cross, the flag of Scotland! (He draws that in blue.) You can also reverse the picture so that the bad set becomes the St. George cross if you prefer to view it that way…..

If $x \in H \backslash N$, we have $\phi (x) = \frac{1}{2} |x - \pi_S (x)|^2 = \frac{1}{2} |\nabla \phi (x)|^2$.

Look at the action (for simplicity $\epsilon = 1$) for a “good curve” $ t \rightarrow x(t)$. Namely a curve for which $x(t) \in H \backslash N$ for a.e. time, the action reads $$ \int_0^t (\frac{1}{2} |\frac{dx}{dt}|^2 + \frac{1}{2} |\nabla \Phi [x(t)]|^2 ) dt. $$

So, obvious minimizers are those good curves that satisfy the first order equation $$ (FO) ~ \frac{dx}{dt} = \nabla \Phi [x] = x - \nabla R [x]. $$

This is a so-called gradient flow of a Lipschitz convex function (up to the first term which can be absorbed). These objects have been studied.

The theory of maximal monotone operators does the job (cf H. Brezis book) in the sense that this is completely well-posed in $H$. We know from that theory that $ x \in C(R_+; H)$, Lipschitz in $t$, and $$ \frac{dx}{dt} (t+0) = x(t) - {d^0 R[x]} $$ which is sometimes called the minimal selection gradient or “mean” gradient (studied in the Italian school).

Example. Differentiate $|x|$. The subgradient fills in the vertical line. The minimal gradient has value zero at $x=0$. This is a nice theory but it gives us very bad curves.

If you start on this St. George cross example, he describes the dynamics and interprets this as a dissipative mechanism. This has little to do with the action principle but it does have dissipation. So, we might take some inspiration from this example….this is a proposal for a modified action.

Modified action:

$$ \int_0^t \frac{1}{2} |\frac{dx}{dt} - d^0 \Phi [x(t)]|^2 dt. $$ Minimizers of the modified action are very likely to be bad curves.

Some rats were confined in a box by electric shocks and another which is very hot. But, if you dig a small channel between the other two boxes. It turns out the rats can survive longer by moving back and forth between the two boxes. I hope it is not a true story….

The dissipation is not incompatible with the arrow of time if you order the data.

Now, I’d like to go back to permutations and fluids. What kid of equation do I get?

Remember the box, broken up into the subcubes. Consider the set $S$ to be the permutations of all the centers. Let $H$ denote $R^{dN}$. In the $d=1$ case, you get a friendly approximate geodesic equation through the classical (nonmodified) action. We are then describing $N$ particles on the line. $$ \epsilon \frac{d^2 x_i}{d t^2} = x_i - \frac{1}{2N} \sum_{j=1}^N ~{\mbox{sgn}} (x_i - x_j). $$ This is like a gravitating parallel pancackes according to Newton gravity plus a repulsive background. This type of model was studied by people like Zeldovich. The repulsive effect is natural in that context. By approximating the incompressible Euler this way, it is nice that you get a model that is reasonable from the point of gravity.

In higher dimensions, the model is NOT consistent with Newtonian gravitation but is instead consistent with a Monge-Ampere correction to Newton’s gravitation. You get something like $\Delta \phi = \rho -1$ and then eventually find something like $ {\mbox{det}} (I + D^2 \phi) = \rho.$ I am not yet certain if this is geometrically reasonable. It is related to Born-Infeld correction to Maxwell’s equations.

So, what is the point? If you modify the action, you can recover interaction with sticky collision.

This is the so-called “Dust” in the Russian literature. These are elementary ideas that explain why matter has clumped in cosmology. Sluggish motions in the early universe moves like honey. Tiny fluctuations of qunatum origin and these create a Jeans instability which tends to concentrate matter. This is at a very large scale and concentrated on a llower dimensional fractal set.

IAS Workshop on Symplectic Dynamics 2: Thursday

Thu, 15 Mar 2012 18:47:33 +0000

IAS School of Mathematics

Workshop web page

Thursday: 2012-03-15

9:00 - 10:00 Peter Topalov, Northeastern University, “Qualitative features of periodic solutions of KdV” abstract
10:15 - 11:15 Jiansheng Geng, Nanjing University, “Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential” abstract
11:30 - 12:30 Massimiliano Berti, UNINA, “Quasi periodic solutions of Hamiltonian PDEs” abstract
2:30 - 3:30 Ralph Saxton, University of New Orleans, “The generalized inviscid Proudman Johnson equation” abstract
4:30 - 5:30 Dongho Chae, Sungkyunkwan University, “On the blow-up problem for the Euler equations and the Liousville type results in the fluid equations” abstract

Peter Topalov: Qualitative features of periodic solutions of KdV

I need to do some detailed setup to expose the ideas I want to describe. We will discuss the KdV equation.

$$ q_t - 6 q q_x + q_{xxx} = 0.$$

Let’s impose periodic boundary conditions. We impose the initial condition $q|{t=0} = q0 \in H^N (T) $. This parameter $N$ will change at different times in the context of the talk, depending upon the theorem we are considering.

$$ H_{KdV} (q) = \int_0^1 (q^3 + \frac{(q_x)^2}{2} ) dx. $$

What is the symplectic (more precisely the Poisson) structure? The phase space where the evolution will happen in $H^N$. For two functions $F,G: H^N \rightarrow R,$ we have the Gardner bracket $$ { F, G } = \int_0^1 \partial_q F \partial_x (\partial_q G) dx. $$ (Ack….I am having trouble making curly brackets show up in the Gardner bracket even when I try to escape using a slash.)

Linearizing around $q=0$, we find $q_t =q_{xxx}$ which we can solve explicitly to find the evolution for the Fourier coefficient: $$ \dot{\hat{q_k}} = - (2 k \pi i)^3 \hat{q_k} = (2 k \pi)^3 i \hat{q_k}. $$ We can solve this directly to find $$ \hat{q_k}(t) = \hat{q_k} e^{i (2k\pi)^3 t}. $$ He draws a collection of complex Fourier planes and draws circles representing the motions of the Fourier coefficients.

Let’s see what the Poisson structure looks like when the dynamics are viewed in terms of the Fourier coefficients.

We compute the Gardner bracket of two Fourier coefficients: $$ { \hat{q_k}, \hat{q_l} } = \int_0^1 e^{-2 k \pi i x} (e^{-2 k \pi i x})’ dx = - 2 l \pi i \delta_{k, -l}. $$ (missing curly brackets on left side.)

We fix attention to zero mean initial data. We will look at $H^N_0$ where the subscript reminds us that we are looking at the zero mean setting.

We define $z_k = \frac{\hat{q_k}}{\sqrt{|k| \pi}}$ and then observe that $z_k = x_k + i y_k$ gives us Darboux coordinates $x_k, y_k$.

We have a mapping $\Phi_L : H^N_0 \rightarrow h^{N+\frac{1}{2}}$. Let’s see why this $\frac{1}{2}$. We take an element of phase space $q$ and apply $\Phi_L$ and this takes us to the associated Darboux coordinates $z_k = \frac{\hat{q_k}}{\sqrt{|k| \pi}}$ and the division by $|k|$ explains the $\frac{1}{2}.$

Remarks about this map $\Phi_L$:

diffeomorphism
canonical
linearizes the flow

Return to KdV.

Theorem 1: $\exists ~ \Phi: H^N_0 \rightarrow h^{N + \frac{1}{2}}$ such that

$\Phi$ is a diffeomorphism;
$\Phi$ is canonical;
$z_k (t) = z_k e^{i \omega_k (q) t}.$
(New) $\Phi = \Phi_L + A; ~\Phi^{-1} = \Phi_L^{-1} + B$ where $A$ is 1-smoothing. What this means is that $A, B$ are bounded maps such that $$A: H^N_0 \rightarrow h^{N + \frac{3}{2}};$$ $$ B: h^{N + \frac{1}{2}} \rightarrow H^{N+1}.$$

In 3. the phases depend only upon the initial data but for some reason I don’t want to write $q_0$ right now.

1., 2., 3. were proven by Kappeler-Poschel-Makarov for $N \geq 0$. For the interval $-1 \leq N \leq 0$, 1.,2.,3. was established by Kappeler-Topalov.

Item 4. is new and recently proven by Kappeler-Schad-Topalov (I didn’t catch the name…). This advance may be viewed as a globalization of a local statement obtained by Kuksin-Perelman.

Consider the KdV evolution moving through phase space. We can also consider the linearized evolution. We are interested in the difference. Denote by $S_t (q)$ the KdV evolution. We can do something a little bit different: $$ S_t (q) - \sum_{k \neq 0} (\hat{q_k} e^{i \omega_k (q) t}) e^{2k \pi i x} = R_t (q). $$

Theorem 2:

$R_t: H^N_0 \rightarrow H^{N+1}_0$ is continuous (even analytic on the Casimir $[q] = 0$).
$\forall ~ q \in H^N_0$, we can consider the orbit $[R_t(q): t \in R] \subset H^{N+1}_0$ is relatively compact.
$\forall ~ M > 0, [R_t (q): t \in R, \| q \|{H^N} \leq M] \subset H^{N+1}0$ is bounded.

In particular, from 2., the norms are relatively bounded.

I want to say something about the proof. The overview involves an expansion of the flow maps using the structure in Theorem 1, item 4. The core of the analysis is in the spectral theory of the Shcrodinger operator.

Jiansheng Geng: Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential

(joint work with J. You an Z. Zhao)

We study a nonlinear Schrodinger equation on the lattice and show there exist quasiperiodic solutions. $$ i \dot{q_n} + \delta( q_{n+1} - q_n) + V_n q_n + |q_n|^2 q_n = 0, n \in Z.$$

Here $\delta $ is small. $V_n (x) = V(n \tilde{\alpha} + x)$ with $V$ a nonconstant real analytic function on R/Z and $\alpha$ satisfying a Diophantine equation.

Eliasson 1997, Acta

Slides moving fast…

Theorem: For small enough $\delta$, this equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions for a.e. $x \in R/Z$.

Also works in the nonlinear case.

Choffrut: What is Whitney smooth? A: Some discussion… Kaloshin: The function is defined on a Cantor set and you need to define what it means to be smooth. You can’t differentiate so you have to do something to understand smoothness….this is the idea of Whitney smooth.

Related works (Linear case):

Belissard-Lima-Scoppola 1983 CMP
Fröhlich-Spencer-Wittwer 1990 CMP
Chulaevsky-Dinaburg 1993 CMP
Eliasson 1997 Acta

Related works (Nonlinear case):

Töplitz-Lipschitz property

Eliasson-Kuksin 2010
Geng-Xu-You 2011

Slides are quite dense, too technical for me to convey here. Abstract KAM theorem.

Massimiliano Berti: Quasi periodic solutions of Hamiltonian PDEs

Nonlinear Wave Equation

$$ (NLW):~ u_{tt} - \Delta u + V(x) u = \epsilon f( \omega t, x, u).$$

$\omega$ diophantine.

Question: Do $\exists$ quasiperiodic solutions of NLW ro $\epsilon \neq 0$?

Linear wave equation: ($\epsilon = 0$.)

Solutions are built by superposition.

Eigenfunctions are orthonormal in $L^2$: “Normal Modes”
Eignevalues $\lambda_j \rightarrow + \infty$: the $\sqrt{\lambda_j}$ are the “Normal frequencies”.

All these linear soutions are periodic. Their superpositions are quasiperiodic. Do these persist when we turn on the nonlinearity.

We look for quasiperiodoc solutions. This leads to an equation for qp solutions:

$$ (\omega \cdot \partial_\phi)^2 u - \Delta u _ V(x)u = f. $$

We can approach this existence question as a bifurcation problem.

We make a NON-RESONANT assumption:

$$ | (\omega \cdot l)^2 - \lambda_j | \geq \frac{\gamma}{1 + |l|^\gamma}, ~ \forall (l,j). $$ The inverse operator is unbounded so the classical implicit function theorem fails. We need a replacement, some kind of Quadratic scheme.

We use a Nash-Moser IFT: Newton method + “smoothing”

The advantage is the rapid convergence. The disadvantage is that we have to invert in a whole neighborhood of the expected solution.

Literature

$d=1$

Kuksin 89, Wayne 90; 2nd order Melnikov non-resonance conditions OK. Dirichlet conditions to ensure simplicity of eignevalues.
Craig-Wayne 93 periodic solutions
Bourgain 94 quasiperiodic solutions

Lyapunov-Schmidt, f analytic, Netwon Method. 1st order Melnikov conditions.

$d \geq 2$

Eigenvalues of $\Delta + V(x)$ appear in clusters of increasing size.
If $d \geq 2$, the eigenfunctions of $-Delta + V(x)$ are NOT localized wrt exponentials! (**Feldman-Knönner-Trubowitz**)

Often, these issues motivate the study of “pseudo-PDEs.”

Newton Method

Bourgain 98 Annals 05 Annals
Wang 10, 11

KAM theory

…Processi, Berti…Craig-Wayne…ack slide changged.

Nash-Moser

Eliasson 89

Berti-Bolle 2011 (to appear in JEMS)

Existence: (Summary of statements; slides are more precise) Under some conditions on $f$, there exists a Cantor like set $C_\epsilon$ of asymptotically full Lebesgue measure. “This is a classical KAM-like statement.”

Regularity:

The Cantor-like set is not technical, e.g. CKSTT 2010 Inventiones.

Pre-assigned direction of tangential frequencies

Geng-Ren 2010
Berti-Biasco CMP 2011
Bambusi-Berti-Magistrelli, JDE 2011

Weaker non-resonance condition

simpler technique

Many of these results should carry over to spheres, Zoll manifolds, Lie groups, homogenous spaces: symmertries and properties of eigenfunctions and eigenvalues are key properties! Related to Birkhoff normal form results by Bambusi, Delort, Grebert, Szeftel for spheres and Zoll manifolds.

For periodic solutions, see Berti-Procesi Duke 2011

Idea of Proof

Small divisors.

Töplitz matrices.

Difficulties:

T has only a polynomial decay off the diagonal.

Smoothing operators; finite-d projectors. TAME estimates are needed. We need estimates on the inverse operator on high regularity Sobolev spaces. Counterexaple of Lojaciewitz-Zehnder! This example shows identifies a parameter boundary in the Newton iteration scheme.

Step 1. $L^2$-estimates: Lower bounds for the eigenvalues.

Step 2. “Separation Properties” of small divisors

Locations where the divisors are small become more and more rare. There emerge “irrational” conditions on the slope $\omega$. These conditions are not needed for the Schrödinger equation. The dispersive relationship is different and helps you here.

KAM

Nash-Moser via the 1st Melnikov conditions. This is in some sense the minimal assumption. This approach works well in case of multiple eigenvalues. However, it has the disadvantage that it requires studying the linearized equation with non-constant coefficients.

Other strategy: impose stronger nonresonant conditions of 2nd Melnikov type (as usual in KAM). This has the advantage that we have a linearized equation with constant coefficients. There exists a torus and a reducible normal form.

Question: Do quasiperiodic solutions persist for nonlinearities which involve derivatives? Important physical applications.

Kuksin 1998
Kappeler-Pöschel 2003
Liu-Yuan 2010 for Hamiltonian DNLS (Benjamin-Ono)

Theorem (Berti-Biasco-Procesi 2011): DNLW has a Cantor-like family of quasiperiodic solutions. These qp solutions have zero Lyapunov exponents and the linearized equations can be reduced to constant coefficients.

Ideas of proof. View this as an infinite dimensional Hamiltonian system. Use conservation of momentum (Geng-You).

Birkhoff Normal form step, reduction to action-angle variables. Then apply an abstract infinite-d KAM theorem.

The Hamiltonian vector field is BOUNDED and “Quasi-Töplitz”.

Procesi-Xu 2011 (introduced Quasi-Töplitz)
Eliasson-Kuksin (similar notion Töplitz-Lipschitz)

Quasi-Töplitz functions

see slides….there is an algebraic closure property of this class under the normal form manipulations.

DNLW

Not Hamiltonian but “reversible” PDE. This is a relaxed setting but which rules out certain nonlinearities like $y_t^3$.

Real coefficients condition which excludes $y_x^3$.

Moser, Arnold, Sevriuk. Algebra of classical reversible KAM theory works out on this PDE as well. The asymptotic expansion of the normal frequencies controlled similarly as in the Hamiltonian case, in analogy with the quasi-Töplitz framework.

Ralph Saxton: The generalized inviscid Proudman Johnson equation

(joint work with Aleajandro Sarria)

This is the Proudman-Johnson (PJ) equation:

$$ (\partial_t + u \partial_x) \partial_x u = \lambda u_x^2 - (\lambda+1) \int_0^1 u_x^2 .$$

This equation comes from the n-dimensional Euler equations. The solutions we consider coming from Euler are unbounded as we go toward spatial infinity so these are infinite energy. He describes some further modeling assumptions culminating into a collapse of Euler into the Proudman-Johnson equation.

History:

Childress, Lerley, Spiegel, Young 1989
Saxton-Tiglay 2008, Okamoto 2009
Okamoto-Zhu 2000
wunsch 2009
A. Constantin 2000

Diverse phenomena as $\lambda$ varies.

Dongho Chae: On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations

Contents

On the blowup problem for Euler
Liouville type equations for fluids

Euler Blowup Problem

Euler 1757

Euler equation on $R^N$.

Kato, Temam, Bouguignon-Brezis. Local existence in $H^m (R^3)$ with $m>5/2$. Do singularities form?

Beale-Kato-Majda 84 Critereon: If there is blowup at time $ {T^*}$ then$$ \int_0^{T^*} \| \omega (s) \|_{L^\infty} ds = \infty. $$
Constantin-Fefferman-Majda 1996 critereon.
Refinements using Triebel-Lizorkhin type spaces. Interpolations.

On the self-similar blowup scenarios:

Self-similar blowup is a popular scenario in search for finte time singularity in nonlinear PDE.
E has a scaling property: $$ v^{\lambda, \alpha} = \lambda^\alpha v(\lambda x, \lambda^{\alpha + 1}t), ~ p = \lambda^{2\alpha} (same).$$

We consider the possibility of self-similar blowups for E.

Energy conservation suggests choosing $\alpha = \frac{N}{2}

Substitute a self-similar ansatz into E to obtain a system called SSE, the self-similar Euler equation. In the Navier-Stokes case, this system is called the Leray system. Leray asked if there exist self-similar blowup solutons for the Navier-Stokes equations in 1930.

Negative answers to Leray’s questions.

$V \in L^3 (R^3)$. Necas-Ruzicka-Sverak 1996
$V \in L^p (R^3), p>3$. Tsai 1998.
Theproofs rely upon maximum principle based arguments, which are not available in the context of the Euler equation.

Theorem (Chae 2007):

Let $V$ be a solution of SSE satisfy

$V \in [C^1 (R^3)]^3$ vanishing near infinity.
There exists $p_1 >0$ such that $\Omega - \nabla \times V \in \bigcap_{0<p<p_1} L^p (R^3).$ Then $V=0.$

The proof of this theorem used the “back to label map” due to Constantin. Recently, I found a much simpler elementary proof.

Chae-Shvydkoy 2012

This is the Euler version of the Navier-Stokes $L^3$ result of Necas-Ruzicka-Sverak.

Nonexistence of asymptotically self-similar blowup Giga-Kohn 1985. See Chae 2007.

Liouville Type Results for Navier-Stokes

Compare with Galdi. Slides are very detailed, provides a survey of the field.

IAS Workshop on Symplectic Dynamics 2: Wednesday

Wed, 14 Mar 2012 18:44:32 +0000

IAS School of Mathematics

Workshop web page

Happy Einstein Birthday!

Wednesday: 2012-03-13

9:00 - 10:00 Wilfrid Gangbo, Georgia Institute of Technology, “Lifting absolutely continuous curves from P(Td) to P2(Rd)” abstract
10:15 - 11:15 Jonatan Lenells, Baylor University, “Geometry of diffeomorphism groupos, complete integrability and optimal transport” abstract
11:30 - 12:30 David Ebin, SUNY, “Groups of diffeomorphisms and geodesics on them” abstract
2:30 - 3:30 Susan Friedlander, University of Southern California, “Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even”. abstract

Wilfrid Gangbo: Lifting absolutely continuous curves from $P(T^d)$ to $P_2(R^d)$

(chalk talk; joint work with A. Tudorascu)

This work extends earlier work on the space of probability measures on the torus $P(T)$ to analogous results on $P(T^d)$. The earlier work used the embedding $P(T) \rightarrow L^2 (0,1)$ but we don’t have this embedding in the higher dimensional case.

Let $P_2 (R^d)$ be the set of Borel measures on $R^d$ with finite second moment $\int |x|^2 \mu (dx) < \infty$. We say $\mu_0 \thicksim \mu_1$ if and only if $\int F d\mu_0 = \int F d \mu_1 ~ \forall F \in C(T^d), ~ \forall F \in C(R^d), ~ \forall F(x+z) = F(x), z \in Z^d$.

I define $P(T) = P(T^d)/\thicksim$.

Let $\gamma$ be a measure on $R^d \times R^d$ which satisfy $\pi_1$ # $ \gamma = \mu_0$ and $\pi_2$ # $ \gamma = \mu_1$. More generally, we write $$W^2_2 (\mu_0, \mu_1) = \inf_\gamma \int_{R^d \times R^d} |x-y|^2 \gamma (dx, dy).$$

Problem: Data: $v:(0,T) \times T^d \rightarrow R^d$ and $t \rightarrow \sigma_t \in P(T^d)$. Assume that $\partial_t \sigma_t + \nabla \cdot (\sigma v) = 0$ (in the sense of distributions). Can we find $t \rightarrow \hat{\sigma_t} \in P_2 (R^d)$ and $\hat{v}: (0,T) \times T^d \rightarrow R^d$ such that $\partial_t \hat{\sigma} + \nabla \cdot (\hat{\sigma} \hat{v}) = 0$ in the sense of distributions. Here $\hat{\sigma_t} \thicksim \sigma_t.$

Such a lift becomes important if I want to associate the rotation number. I want to write $$ \frac{d}{dt} \int_{T^d} x d \sigma_t = \int_{T^d} v_t d \sigma_t$. $$

Weak KAM: (A small fraction of what is known) $M= T^d$. Let $h: T^* M \rightarrow R$. Let $w(z_0, z_1) = z_0 (J z_1)$ where $J$ is the usual matrix satisfying $J^2 = - Id$. Let $X_h$ denote the associated Hamiltonian vector field: $$ \dot{\phi} = X_h (\phi), \phi_0 = (x_0, p_0). $$ The associated flow is denoted $\phi_t = (x_t, p_t).$

Existence of weak Lagrangian Tori: $\overline{h}: R^d \rightarrow R$ is the effective Hamiltonian, $ c \in R^d$.

$\exists ~ u \in C(t^d)$ with $h(x, c+ \nabla u) = \overline{h} (c)$ (viscosity)
$u_c (x) = c \cdot x + u(t).$ ($\partial u_c$ is invariant under $\phi$.)
$\forall ~ x_0 \in T^d ~ \exists v_0 $ such that if $(x_t, p_t) = \phi_t$ then $\forall ~ T$ $$ u(x_T) - u(x_0) = \int_0^T [l(x,\dot{x}) + c \cdot \dot{x} + \overline{h} (c)] dt$$ $$ \lim_{t \rightarrow \infty} \frac{\hat{x_t}}{t} = - \nabla \overline{h} (c). $$

General Fact: $M \rightarrow $ compact.

Specific to finite $d$:

Given $x \in W^{1,2} (0,T, T^d)$ and take two lifts $\hat{x}, \hat{y} \in W^{1,2, R^d}$. We then find that $$ \hat{x_t} - \hat{y_t} = n \in Z^d $$ because we have $$ \lim_{t \rightarrow \infty} \frac{\hat{x_t}}{t} = \lim_{t \rightarrow \infty} \frac{\hat{y_t}}{t} $$

Obstacle in infinite dimensions:

Let $M_0 = P(T^d)$ and in the sense of distributions we have $$\partial_t \sigma + \nabla \cdot (\sigma v) = 0.$$

If $\nabla \cdot (\sigma w) = 0$ then $v+w$ is another velocity.

(wash board…)

Let $\mu \in P_2 (R^d)$ and define the tangent space $T_\mu P_2 (R^d)$ and also the space $T_\mu P(T^d)$. These are defined with $L^2$ closures.

Pseudo symplectic form: ….going faster and I’m not keeping up with the typing.

Theorem (Gangbo-Kun-Pacuni 2011):

$\Omega$ is a closed skew symmetric nondegenerate 2-form.
$\exists ~ X_H$ such that $ -dH = \Omega (X_H, \cdot)$.
$\dot{f} = X_H (f) \iff \partial_t f + \nabla_x (vf) = \nabla_v (f (\nabla V + \nabla W * \rho)).$

This is a nonlinear Vlasov equation.

I want to state the analog of the weak KAM theorem in our context.

Theorem Let $\overline{H}$ be the effective Hamiltonian of $H$ restricted to $R^d$ and let $c \in R^d$.

$\exists ~ U: P(T^d) \rightarrow R$ such that (in viscosity sense) $$ H(\mu, c + \nabla_w H) = \overline{H} (c). $$
Given $\sigma_0 \in P(T^d) ~ \exists ~ v_0: T^d \rightarrow R^d$ such that if $f_0 = \sigma_0 \delta_{{v0}}$ then $$ f_t = \sigma_t \delta_{{vt}}, $$ $$ U(\sigma_t) - U(\sigma_0) = \int_0^T [ L(\sigma_t, v_t) + \int_{R^d} v_t \cdot c d\sigma_t + \overline{H} (c)] dt. $$
We also have $$ | \frac{1}{T} \int_0^T dt \int_{R^d} v_t d\sigma_t +\nabla \overline{H} (c) | \leq \frac{const}{\sqrt{T}}. $$

Corollary: If $(\hat{\sigma}, \hat{v_t})$ is an appropriate lift then $$\lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T ( \int_{R^d} x d\sigma_t) dt = - \nabla \overline{H} (c).$$

Questions:

Mather: This has connections with fluids?

Answer: Kinetic theory. Consider the system $ \partial_t^2 x = \frac{1}{N} \sum_{j=1}^N W(x_i - x_j) - \nabla V(x_i)$. When we consider the $N \rightarrow \infty$ limit, we can move the weak KAM theory from this $N$ particle system to the infinite particle case by moving to the setting of measures. This framework lets us prove convergence of discrete models to the PDE case.

Jonatan Lenells: Geometry of diffeomorphism groups, complete integrability and optimal transport

(pdf slides; Happy $\Pi$ day!; Einstein’s birthday)

(joint work with B. Khesin, G. Misiolek, S. Preston)

Outline

A new equation
Geometry of $Diff(M)$
A sphere
Optimal Transport
Geometric Statistics

A new equation

$$ \rho_t + u \cdot \nabla \rho + \frac{1}{2} \rho^2 = \frac{- \int_M \rho^2 d \mu}{2 \mu(M)}. $$

$M$ is a compact Riemannian manifold.
$\mu(M)$ is the volume of $M$.
This is an exciting equation because it is completely integrable for any $M$.
This is a geodesic equation on $Diff(M)/Diff_\mu (M)$.
Describes $\dot{H}^1$-optimal transport
reduces to the Hunter-Saxton equation for $M=S^1$. (derived in the context of liquid crystals in the early 90s.)

Euler-Arnold Equations. Summary of those ideas.

abc-metric. You can add some other terms to the original $L^2$ inner product involving $L^2$ inner products involvling codifferentials and the musical isomorphism. A lot of different equations arise as you take different values of the parameters. Writing down the associated abc Euler-Arnold equation generates a big equation which can be specialized into various equations. To obtain the $\dot{H}^1$ metric, we cancel away the terms associated with factors a and c. We simplify by setting $a=0, b= \frac{1}{4}, c = 0$.

The equation induced by these choices has some degeneracy issues. These can be resolved by quotienting out part of the phase space. The function $u$ is not uniquely determined but its coset is uniquely determined. (Similar issues arise in Hunter-Saxton.) The equation we are considering here is a geodesic equation on a (quotiented) Diffeomorphism group.

Jacobian determinant.

A sphere

Theorem (Khesin-Misiolek-Lennels-Preston): The map which takes the coset $[\eta]$ to its associated Jacoobian $\sqrt{Jac_\mu \eta}$ is an isometry onto a subset of the sphere.

This isometry lets them transport all the questions about the geodesic equation on this complicated Diff phase space into corresponding questions about geodesics on the sphere. Since we understand the sphere well, we can conjugate results there using the mapping to obtain explicit solution formulae for the geodesic equation. Magical integrability!

Preceding works.

Khesin-Misiolek 2003: Showed Hunter-Saxton may be viewed within the Euler-Arnold framework.

Lennels 2006: Recognized the image of the map as a portion of the sphere.

Optimal Transport

Optimal Transport. Wasserstein distance between two probability measures.

Moser 1965, Ebin-Marsden 1970, Otto 2001, also Benamou-Brenier.

The $\dot{H}^1$ optimal distance induces what they call the spherical Hellinger distance since it resembles the Hellinger distance used in probability theory

Geometric Statistics

Statistical model.

Fisher-Rao information metric.

Theorem (KMLP): The $\dot{H}^1$ metric coincides with the Fisher-Rao metric when restricted to any k-dimensional submanifold of the (quotiented) $Diff$.

This is another reason why we think this metric is important. It arises from many different points of view.

Summary

We found a new integrable PDE.
The PDE is a geodesic equation on a quotieted diff with $\dot{H}^1$ metric.
A sphere
One can understand what is going on using the optimal transportation point of view using the $\dot{H}^1$ metric.
This metric coincides with a basic metric arising in geometric statistics.

Open problems

Global weak solutions of the PDE? All solutions break in finite time because you hi the boundary of the diffeomorphism coset. However, there is no problem when you view the dynamics on the sphere. The motion along the great circle may be continued. This type of development has taken place already in the context of the Hunter-Saxton equation. The process appears to be more complicated in this more general context since the Jacobian can vanish.
Transfer results from geometric statistics into this diffeomorphism quotient. Then reinterpret these objects in the setting of PDE. Amari-Nagaoka 2000 alpha-connections, dual connections.
Develop an optimal transport theory based on the $\dot{H}^1$ theory.
Find a Lax pair.
Find a bi-Hamiltonian structure.
Analyze the associated two-component equation (c.f. Lennels-Zhao 2011). There is a 2-component Hunter-Saxton so that object suggests we might find a corresponding generalization. This has been observed by LZ.

David Ebin: Groups of diffeomorphisms and geodesics on them

(joint work with Stephen Preston)

Maps from a manifold to itself. Discussion of various topologies of such maps.

Volume preserving maps. Diffeomorphisms (Volumorphisms)
Even dimensional manifolds with a symplectic form. We can consider the maps which preserve the symplectic form. (Symplectomorphisms)
For odd dimensional manifolds, we can consider maps which preserve the contact form. (Contactomorphisms)

In all these cases, we can discuss the geodesics…..ack low battery.

Boothby-Wang fibration.

Susan Friedlander: Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even

(joint work with Vlad Vicol, Walter Rusin, Francisco Gancedo, Weiran Sun)

Homage to Oscar Wilde…

Amain theme is that there is a difference in behavior of solutions in Active Scalar Equations when the associated Fourier multiplier is even versus odd.

Active Scalar Equations; Incompressible Fluids

$$\theta_t + u \cdot \nabla \theta =0. $$ $$ \nabla \cdot u = 0. $$

$$ u = O [\theta], ~ PDO $$

$R^d$ or $T^d$. Even or odd Fourier multiplier symbol. The results I’ll describe are not influenced by the presence of a physical boundary. The emphasis will be on examining the influence of the operator O on the properties of the PDE.

Consider $u_j = \partial_i T_{ij} \theta, ~ \nabla \cdot u = $. Here $T_{ij}$ is a $d \times d$ Calderon-Zygmund operators.

ODD Symbol: Locally well-posed in Sobolev spaces. Commutator in energy estimates. **Chae et. al, Friedlander-Vicol.
EVEN Symbol: Lipschitz ill-posed in Sobolev spaces. Friedlander-Vicol; Nonuniqueness for $L^\infty$-weak solutions. Techniques from convext integration. Shuydkoy.

Recent reviews of results for certain active scalar equations. “Regularity and blowup for active scalars” Kiselev 2010.

SQG Equation: $u = R^\perp \theta$, symbol $\frac{i (k_2, -k_1)}{|k|}$.

Constantin-Majda-Tabak 1994, Resnick 1995 (Chicago thesis; unpublished), Wu, Cordoba, Chae, Iyer, Ju, Fefferman

The SQG equation had been known in the geophysics community before its introduction to the mathematical community by Constantin et.al.

Local existence for smooth initial data BUT global existence of smooth solutions is OPEN (just as it is open for 3D Euler). Cordoba and Fefferman have ruled out the existence of certain solution scenarios.

“Modified” SQG Equation (Okhitani)

Insert a power of $(-\Delta)^{1/2} = \Lambda$ in the map $\theta \rightarrow u$ so that $$ u = \nabla^\perp \Lambda^{\beta-2} \theta $$ where $1 < \beta \leq 2. Chae, Constantin, Cordoba, Ganceda, Wu 2011. Local existence of smooth solutions in $H^s$, global existence of weak solutions.

Note: result holds more generally when the symbol is ODD and order $\leq 1$.

IPM equation: singular integral operator with EVEN symbol

Darcy’s law.

$$ u = R^\perp R_1 \theta. $$

Cordoba-Gancedo-Orive 2007

Regular initial data, local existence, weak solutions, SQG and IPM present different behaviours. Global existence of smooth solutions is OPEN.

Even symbol: $$( \frac{k_1 k_2}{|k|^2}, \frac{-k_1^2}{|k|^2}).$$

There is very different behaviors among these equations for rough data. “Patch type initial data”

SIPM equations (even, unbounded)

$$ u = R^\perp R_1 \Lambda^\beta \theta$$

Friedlander-Gancedo-Sun-Vicol (2012)

Locally Lipschitz ill-posed in $H^s, ~ s>2$. Proved for $0 < \beta \leq 2$ in $T^d \times [0,\infty]$
Locally well-posed for some “patch-type” weak solutions. Proved for $0<\beta <1 $ in $R^2$.

(Discussion: the notions of “wellposedness” changes between the previous two bullet points.)

Symbol: $k_1 k^\perp |k|^{\beta -2}$

Magnetogeostrophic (MG) equations

Friedlander-Vicol 2011

Long symbol $M$, even, unbounded, 3D. $ u = M\theta$. Here $M$ is a vector operator that defines a 3-vector $u$.

Cauchy problem is ill-posed in Hadamard sense in Sobolev spaces. There is no Lipschitz solution map.

Ill-posedness: singular, even, symbol

Active scalar equation. Special direction with index $d$, often associated with gravity. A list of many conditions on the $d$th component $S_d$ of the Fourier multiplier operator….allowing them to build eigenfunctions to show Lipschitz failure.

Definition: Locally Lipschitz $(X,Y)$ well-posed.

$$| \theta_1 (\cdot, t) - \theta_1 (\cdot, t)|X \leq K | \theta1 (\cdot, 0) -\theta_2 (\cdot, 0) |_Y. $$

The spaces $X,Y$ are often chosen to be $(H^r, H^S)$.

Theorem: Under the many assumptions on $S_d$, the active scalar equation is Lipschitz $(H^r, H^s)$-illposed for any $r > R, s \geq r+1$.

Linear problem

Linearize around $\theta_0 = \sin m x_d $. Write out a Fourier series. Crank out a recurrence relation.

Continued fractions, characteristic equation. These ideas where used by Michalkin and Sinai to show unstable eigenvalues for the shear flow for Navier-Stokes equations.

Ill-posedness of the nonlinear problem

Follows a proof by contradiction.

Effects of dissipation: MG

Dissipation: $ \nu (-Delta)^{1/2}$

Using De Giorgi techniques, Caffarelli-Vasseur proved critical SQG. What can we say about the MG equation?

Case $1/2 M \gamma < 1$: LWP in $H^s$, for $ s> \frac{5}{2} + (1 - 2\gamma)$. Well-prepared initial data.
Case $ 0 < \gamma < 1/2$: Diffusion is too weak to overcome the continued fraction construction.
Case $\gamma = 1/2$: Unique global solution when the initial data and source are small in a suitable sense, then there exists a unique golbal solution. However, if the data are large in this respect then we can run the ill-posedness construction. This reveals a very precise dichotomy.

IAS Workshop on Symplectic Dynamics 2: Tuesday

Tue, 13 Mar 2012 18:42:10 +0000

IAS School of Mathematics

Workshop web page

Tuesday: 2012-03-13

9:00 - 10:00 Laszlo Szekelyhidi, University of Leipzig, “The h-principle for the Euler equations” abstract
10:15 - 11:15 Camilo de Lellis, University of Zurich, “The h-principle for the Euler equations Part 2” abstract
11:30 - 12:30 Vladimir Sverak, University of Minnesota, “On the long-time dynamics of some infinite-dimensional Hamiltonian systems” abstract
2:30 - 3:30 Antoine Choffrut, University of Leipzig, “On the local structure of the set of stationary flows to the 2D incompressible Euler equations” abstract
4:30 - 5:30 Thomas Kappeler, University of Zurich, “Symplectic techniques for integrable PDEs” abstract

Laszlo Szekelyhidi: The h-principle for the Euler equations

(chalk talk; joint work w. Camillo De Lellis)

“What I will speak about has nothing to do with symplectic and nothing to do with dynamics.” Hofer: very good.

Euler

$$\partial_t v + \nabla \cdot (v \otimes v) + \nabla p = 0; \nabla \cdot v = 0.$$

The spatial dimension $n=2,3$, certainly $>1$. We will speak about weak solutions $v \in L^2_{loc} (T^n \times [0,T])$ and we throw all derivatives onto test functions.

Why look at weak solutions? The equations tell you conservation of mass and momentum. The derivation is done from a continuum analysis so this formulation is natural. For $n=3$, another reason is the relationship with turbulence (K41, O49). The story starts with anomalous dissiipation. The observation is that if you consider $NS_\nu$ with small $\nu$ then formally, the dissipation rate $$ \nu \int |\nabla v|^2 dx > \epsilon. $$ This is observed in experiments as $\nu \rightarrow 0$. If you plot $\log k$ vs. $\log E(k)$ there are three different regimes: a low frequency regime related to the geometry of the domain, an inertial range with slope $-5/3$ and then a rapid dissipation at high frequencies. Bob Kohn once said that a log-log plot always looks linear. ….discussion with Peter and Camillo….”Bob Kohn is not here so let’s leave him alone.” K41 looked at ensemble averages and he derived the $-5/3$ scaling law. O49 was thinking of a single solution. He said that if we beleive in this kind of log-log picture then in the intermediate regime we are far from dissipation so the nonlinear term is responsible for the $-5/3$ decay. Is it possible to see a single solution that displays this type of decay. If you translate this point of view into a regularity statement and look for $ v \in L^\infty_t C^\alpha_x$ then when $\alpha > 1/3$ we have energy conservation and for $\alpha < 1/3$ then you have anomalous dissipation possible. Klainerman: Is this $1/3$ easy to see? Discussion: Yes, it is just scaling, look at Fourier coefficients….

Eyink, Constantin-E-Titi solved the $\alpha > 1/3$ part of Onsager’s conjecture. There is basically nothing known in the $\alpha < 1/3$.

Spencer: Uniqueness in that range? Answer: No you need Lipschitz to see uniqueness so there remains a big gap.

Studying weak solutions puts us ina different framework than the study of smooth solutions, long time behavior for 2D,etc. This is a different world.

Theorem (Scheffer-Shnirelman): There exists a nontrivial weak solution with compact support in time.

This solution can be thought of as having initial data zero, then it is not zero and after a while, it is zero again. (This solution is far from regularity $1/3$.)

h-principle (Gromov)

This theorem can be viewed as a statement of the form of the h-principle. This principle should be viewed as a different tpe of statement related to Hadamard ill-posedness.

Theorem (Nash-Kuiper): Any strictly short smooth embedding of (compact) $M^n \rightarrow R^{n+1}$ can be uniformly approximated by $c^1$ isometric embeddings.

For a geometer, this is viewed as a completely wrong theorem. It seemingly contradicts the classical rigidity of the 2-sphere. Any isometric embedding of the 2-sphere into $R^3$ is the standard embedding. However, that theorem requires curvature so needs $C^2$.

Berti: What is short? Answer: Distances in the image are shorter than distances in the domain. So, Lipschitz with constant less than 1.

Two conditions: A topological global condition, an embedding. A local condition, isometric.

global: embedding
local: isometric

General statement. If you can satisfy the global constraint, you can twist it satisfy the local statement. Another example is Gromov’s theorem saying 2 forms can be converted into symplectic forms.

An idea of the proof of this statement (Nash): $M^n \rightarrow R^{n+2}$. This is basically about a single chart so let’s look instead at $\Omega \subset R^n$ and we consider the embedding $\Omega \rightarrow R^{n+2}$. We have $$ \nabla u^T \nabla u = g $$ and strictly short means $g - \nabla u^T \nabla u >0$. We can make wrinkles. Wrinkling is written as a spiral $$\tilde{u} (x) = u(x) + \frac{a(x)}{\lambda} (\sin (\lambda x \cdot \xi) \zeta (x) + \cos (\lambda x \cdot \xi) \eta (x)),$$ where $\zeta, \eta$ are unit normal to $u(\omega)$. This looks like a “telephone cord”. What happens in orthogonal directions? How does this affect the metric? This is something you can calculate: $$ \nabla \tilde{u}^T \nabla \tilde{u} = \nabla u^T \nabla u + a^2 (x) \zeta \otimes \zeta + O(\frac{1}{\lambda}). $$ This means I have a lot of freedom to change the metric in a fixed given direction. This allows me to write $g - \nabla u^T \nabla u $ as a sum of terms $\sum a_j^2 (x) \xi^j \otimes \xi^j.$ It is important here that the $\xi$ does not depend upon $x$. This allows me to achieve a reduction in the difference

$$| g - \nabla u^T \nabla u |_0 = O(\frac{1}{\lambda}),$$

$$| u - \tilde{u} |_0 = O(\frac{1}{\lambda}), $$

$$| u - \tilde{u} |1 \thicksim | a |0 \thicksim | g - \nabla u^T \nabla u |_0^{1/2}.$$

Lipschitz isometries

$\Omega \rightarrow R^n$

Kirchlein Baire Category Method.

Consider the space $X = [ u \in Lip (\Omega): \nabla u^T \nabla u \leq Id]$ endowed with the supremum norm. The observation of Kirchlein is that $\nabla \cdot X \rightarrow L^1$ is Baire-1. Consider 1-Libschitz maps converging to the zero function. With this same argument, I can take any function and add corrugations.

Mather: What is Baire-1? Answer: It is a pointwise limit of continuous maps.

A corollary of Baire Category theorem. The points of continuity is dense. Despite the troubles with the corrugation possibility, most maps in this space are points of continuity. The only places where you can’t improve is where the gradient is already maximizing. As a consequence, most maps in this space are isometric.

This is an argument which can be generalized quite a bit and can be applied to the Euler equations.

Example. The original system (O). (Tartar-DiPerna ideas)

$$\sum_{i=1}^n A_i \partial_i z = 0, ~in D’$$ $$ z(x) \in K, ~a.e. ~x$$

and we want to move to same relaxed condition (R) but with $z(x) \in K^{\Lambda}$, which he refers to as the convex hull of $K$. The principle is that most (in the sense of Baire Category) solutions of the relaxed setting R are solutions of the original problem O.

Question: What is the set $K^\Lambda$? This is the “wave cone”. Let $z: R^n \rightarrow R^d$ here. $\Lambda = [ \hat{z}: \exists \xi \in S^{n-1} ~s.t.~ \sum_i A_i \xi_i \hat{z} = 0]$. So, these are the directions in which we can oscillate while maintaining the conservation law and keeping the constitutive relations intact. So, that is $\Lambda$ and then the $\Lambda$-convex hull is like this. $z \notin K^\Lambda$ if $\exists ~ f ~ \Lambda$-convex so that $f(z) > 0, f|_K \leq 0$. You can separate. This general point of view was developed by Tartar, DiPerna.

Klainerman: what is the “h”? Answer: In this setting “h” stands for homotopy but that is not so present in this discussion. My view of the weak version of the h-principle is that there are a lot of solutions which have less regularity. De Lellis: Gromov woud say that you can take your short map and homotopize it while maintaining that it is an isometry, except at the endpoint.

Application to Euler

Now, you can write the Euler equations in this form by renaming the nonlinearity as a new variable. He shows how to do this by renaming some variables, interprets the associated $K$ and $K^\Lambda$.

Theorem (DL-Sz): Let $\overline{e}$ be a given function on $T^n \times [0,T]$ and let $(\overline{v}, \overline{u}, \overline{q})$ be smooth strict subsolution. Then $\exists ~ v_k \in L^\infty$, a sequence of weak solutions of Euler such that $v_k \rightharpoonup \overline{v}$ in $L^\infty$ and $\frac{|v_k|^2}{2} = \overline{e}$ a.e. $(x,t)$.

This is the “local part” of the h-principle. Given one subsolution, I can construct a solution by adding these waves. More or less, what Scheffer-Shnirelman have done is to take 0 as the subsolution.

….I am very much running out of time….so just to state one more theorem which touches the initial data.

Admissibility. Those weak solutions for which the $L^2$ norm is nonincreasing. Under this type of assumption, you have the weak-strong uniqueness. Any such strong solution is unique within the larger class of weak solutions emerging from the same initial data.

Theorem: Let $n=2$. Let $v_0(x)$ be the shear flow (he indicates this graphically wiht an interface and an arrow to the right above and an arrow to left below). $\exists~$ infinitely many admissible weak solutions.

He draws the interface with a thickened interface of some growing-with-time size outside of which we have the share flow.

Among all the selection criteria you might be considering for restoring uniqueness among the weak solutions, you could ask for maximally dissipating, you could choose the shear flow itself. Or you could ask for the one which has the fastest interface thickening. It is not yet clear which is the physically relevant selection critereon.

Questions:

Sverak: If you take a sequence of smooth solutions onverging to your data, is there any relation to your solution? Answer: It depends how it converges. Sverak: The best you can with continuous. Answer: I’m not sure.

Camilo de Lellis: The h-principle for the Euler equations Part 2

(continuation of previous talk; chalk talk)

I’ll start by mentioning some related results in the literature.

Survey article: D-Sz posted on web in 2011 contains all this literature.

Wiedeman: Global existence of weak solutions for any $L^2$ initial data in $R^3$. This would have been a fantastic theorem if it had not been too many solutions. This is the global analog of what was outlined before. Kappeler: ARe they adminssible? Answer: No. We are not anywhere near the blowup problem.
Sz-Wiedemann: You can approximate any measure-valued solutions (a la DiPerna-Majda) with exact solutions.
Sz-Wiedemann: The set of “bad” initial data is $L^2$-dense.

These techniques can also be applied to other equations.

Incompressible porous medium equations. Some class of active scalar equations. Caddoba-Faoco-Grancedo, Shydkoy, Sz.
Compressible Euler. D-Sz, Chiararoli, Chiadaroli-D. (Higher dimensional conservation laws have a striking nonuniqueness, contrast with the entropy conditions in 1d)

The fact that there exist $C^1$ isometric embeddings uniformly approximating any short smooth embedding was surprising.

Theorem (D-Sz 2011): For any given $e: [0,1] \rightarrow R^+$ smooth. Then there exists a $C^0$ solution of incompressible Euler in $T^3 \times [0,1]$ such that $$ e(t) = \frac{1}{2} \int \frac{|v|^2}{2} (x,t) dx. $$

We are moving towards the lower part of the conjecture of Onsager. I can’t claim this is saying anything about turbulence, but it does speak to the issues of dynamics of solutions viewed on Fourier coefficients.

Remark: It seems we can reach some Holder regularity, something explicit like $\frac{1}{500}$.

Shnirelman: Can you say something about modulus of regularity about this solution? Answer: You can work out something. It would be painful to work out. Finding a Holder exponent is achieved through an iteration process.

The process uses smoothness properties of $e(\cdot)$.

….change gears.

Borisov ‘50: If $v \in C^{1,\frac{2}{3} + \epsilon}$ is an isometric embedding of a positively curved connected 2d surface in $R^3$ then the image is convex.

This gives you rigidity. This is a local theorem. The global theorem would say that the sphere has an isometric rigidity. Borisov also had an announcement…..never published his proof.

Borisov (1965—>2004): h-principle for 2d analytic surfaces in $R^3$ if $u \in C^{0, \frac{1}{13} - \epsilon}$.

Conti; D-Sz: Rigidity and h-principle in general dimension (with better exponenets). For 2d the exponent is $\frac{1}{7}$.

He describes a double iteration scheme. He emphasizes that the parameter $\xi_0$ appearing inside the $\sin$ and $\cos$ is independent of $x$.

There are always successive one dimensional layers in any convex integration, in any h-principle appliation. To improve these constructions and obtain the $C^0$ statement we need to replace these constructions by higher dimensional generalizations. This is a direction suggested also by Gromov in his book.

…slowing down on typing….I’m just going to watch this.

Vladimir Sverak: On the long-time dynamics of some infinite-dimensional Hamiltonian systems

2 main examples

$w_t + u \nabla w =0, ~ w = \curl u, \nabla \cdot u = 0, x \in T^2
$NLS_3 (T^2)$

Thanks to Shnirelman, Kuksin, Choffrut, Keel, Polacik (slide changed fast might be incomplete).

Speculative picutre in Fourier space.

There is an interesting possibilty that on the macroscopc scale the dynamics might be simpler than in finite dimensional cases. Why? If you look at the dynamics on the Fourier space, at time $t=0$, we impose some nice initial data and as tiem evolves, some part of the solution moves toward infinite frequency. Everything in this direction is very hard to establish. There are some obstructions to this behavior. For example, KAM and complete integrability block this phenomena. We consider here “generic solutions”. Even though the initial data might be very complicated, we might end up with complexity moving into the high Fourier modes and what remains will be dictated by the conservation laws. This should be contrasted with finite-d systems. If one truncates the PDE in a naive way, we will initially see the same type of picture (motion toward infinite frequencies) and then we will hit the frequency cutoff and there will eventually be a thermalization. The “complexity” has “nowhere to go”.

Let’s look at defocusing $NLS_3$. We have, for this equation, three basic conserved quantities:

Energy
Momentum
Mass

Variational principal. We can think of the equation as a variatonal principal: minimize E subject to the mass=m and momentum=p constraints. We can compare this situation to 2D Euler. (In 3D there is the possibility that everything goes to infinity.) The $L^2$ norm and momentum provide constraints and the minimization principle gives you some nontrivial solutions even in the linear case.

The most optimisitic scenario for the transfer to high frequencies is that for a generic solution over a long period of time, the solution will spend most of its time (in some weak topology) near this manifold of minimizers of $E$ subject to the constraints given by parameters $p,m$. The natural “phase space” for solutions is $H^1$ and we view the manifold $M(p,m) subset H^1$. Ergodicity.

The variational principle may be viewed from the statistical mechanics point of view. Consider a finite-d truncation via Dirichlet prjection. If we believe in stat mech in this scenario, we can use the microcanonical ensemble and look at the set of all points in our phase space where our energy lies between $E$ and $E + \delta$. We have a natural volume measure on this space. We can similarly $\delta$-thicken around the momentum and mass level sets. We can then hope that the solution will concentrate onto a measure living on these subspaces. This is provably rigorous in the linear case. In the limit as the truncation parameter goes to infinity, the energy $E$ is “forgotten” while the mass and momentum constraints are “remembered”. Another way to look at it is to follow the rule-of-thumb that the energy is equidistributed across possible states. Our initial energy is finite and we are equidistributing it across more and more states. Therefore, the temperature (energy per state) and the whole solution will weakly converge to zero. So, as $N \rightarrow \infty$, we concentrate on low frequencies consistent with the constraints and the rest of the solution goes to zero temperature.

Comparison with Gibbs measure. (Lebowitz, Bourgain,…) In this construction, one exponentiates the Hamiltonian and interprets this as a density with respect to the Wiener measure. For the Gibbs measure, $\langle E \rangle = + \infty$ and $\beta > 0$. These functions live on function spaces with infinite energy. The measure is concentrated on functions with low regularity and infinite energy. Remarkably, the dynamics is still well-defined by the PDE (Bourgain).

A rigorous result. NLS defines a dynamical system on the space $(X, w)$, where $X = [ \psi \in H^1, | \psi |_{H^1}\leq C]$. Here “w” denotes the weak topology. This is OK because Bourgain has shown well-posedness on $H^s, s<1$ (Bourgain).

Theorem: The $\Omega$-limit set (wrt the weak topology) contains solutions for which no movement to high frequencies goes to infinity. These are precompact in $H^1$.

These are the “end states” which solutions approach in the weak topology.

Question: Does every $\psi \in \Omega_+ (\psi_0)$ have this property?

Heuristics: If $\psi \in \Omega_+ (\psi_0)$ then the high frequency part has already separated.

Interpretation:

The solutions do not quite approach $M(p,m)$. there is probably some escape….ack slide changed.

2D incompressible Euler

the situation is quite similar. The difference between 2D Euler and NLS is that, in some sense, 2D Euler is a Poisson system instead of a Hamiltonian system. The system is a Hamiltonian system on symplectic leaves. Formally, 2D Euler should be viewed as a family of Hamiltonian systems and the orbit takes place on the leaf.

Fourier representation. Stream function. Energy. Conserved quantities associated with vorticity transport.

Natural “phase space”…. $L^\infty (T^2)$ with a weak * topology. Yudovich has shown the evolution is well-posed in $L^\infty (T^2)$. We have the necessary well-posedness pieces in place. Importantly for us, in this context of developing fine structures in teh flow, we have the stability result: If the data converges weak start then the same is true at later times. this follows from the proof of Yudovich.

Analogy with NLS. He draws a table.

Phase space: $H^1$ …. $L^\infty$
Weakly continuous: Momentum, mass …. $ E = \int -\frac{1}{2} \omega.$
Lower semicontinues: Energy….. ack slide changed.

Fourier picture. Variational principle, related to the notion of “mixing” introduced by A. Shnirelman). We minimize the energy subject to the constraints associated to the weakly continuous quantities. This produces a steady state solution (whih depends on $f_0$).

More geometric picture. We know that a good topology is the weak * topology on $L^\infty$. We can therefore study the weak * closure of the orbit.

Example: Data that looks like $\omega_0 = \chi_A - \chi_B.$ Here $B = T^2 \backslash A, ~ |A| = |B|$.

Onsager 1947, Montgomery-Johce, 1970s, Miller, Robert 1990s, Turkington 1990s, closely related to Shnirelman’s notion of “mixing”.

Looks similar to Ising model, except that the interaction is long-range. “Most-probable” configuration for a given energy $E$? But we specify here the number of configuration cells that have sign +1, and how many have -1. We can then do the usual statistical mechanics calculations.

He draws another analogy diagram.

NLS …. 2D Euler
$C^N$…..Ising config
Classical Maxwell Boltzmann microcannical ensemble picture……fermions (generalized) with long range interaction.

Full ergodicity seems….ack.

A more geometric picture for Euler (a sketch).

Geometric finite-d approximations inside the $SDiff (T^2)$ viewpoint. Remarkable fact learned from Khesin book showing that we have finite-d geometric approximations using $SU(N)$.

Determining the “end states” for Euler.

Calcuating the “entropy” etc., one gets different answers depending upon how one counts.

In Euler on the torus, the temperature is not zero but is instead negative as was observed by Onsager. All these predictions suggest that hte “end states” consist of shear flows.

An example where transfer to high frequencies was proved: Landau damping.

The only situation where this was rigorously proved wiath the Vlasov-Poisson system.

Landau 1946
Caglioti-Maffei
Hwang-Velazquez 2008
Mouhot-Villani 2009

Is there an analog of Landau dampoing possible for 2D Euler?

Linear Landau damping: Yes (Sverak)
Nonlinear case: probably yes, but seems more difficult than the Vlasov-Poisson case.

Is there an analog for this in the 2d NLS case? This seems more difficult in the NLS case. Transfer to high frequencies for dispersive equations:

2d NLS: “I-team”
Various model situations: Bourgain,…

Questions: 1. Can you formulate a stability statement related to your question about the “end states” in the Schrodinger setting? Answer: I expect there should be stability statements around the energy minimizers subject to mass and momentum constraints. 2. Do you expect corresponding statements for the focusing problem (when you don’t expect blowup)? Yes, but there will be a weak convergence to soliton instead of the “breather” energy minimizers.

In discussion after the talk, Sverak suggested that these minimizers are not localized onto single Fourier modes and instead appear to be some kind of “breather” solution. These objects should have variational stability properties resembling corresponding statements about solitons and built along the motif of Arnold Stability results as appearing in the book by Khesin.

T. Oh reported to me that some studies like those suggested in this talk appear in work of Chatterjee-Kirkpatrick.

Antoine Choffrut: On the local structure of the set of stationary flows to the 2D incompressible Euler equations

arXiv

(chalk talk)

The Euler flow evolves on symplectic leaves. If you start on one of these leaves, then the Euler flow stays on the leaf. The result I want to prevent is the following. Suppose you have a stationary solution on one leaf, then the other stationary solutions are located on a curve that passes through the leaves. There is a 1-1 correspondence between the stationary solutions and the leaves.

This talk will be a bit more elementary than the other talks. Some aspects were forecasted by Preston and Sverak in earlier talks in this workshop.

Consider a (potato shaped) domain $\Omega$. The Euler equation $ \partial_t u + (u \cdot \nabla) u _ \nabla p =0, \nabla \cdot u =0, u \cdot N =0 (\partial \Omega).$

In 2D, we introduce $u = (u^1, u^2)$ and the vorticity $w =\partial_x u^2 - \partial_y u^1$ and we can derive the vorticity equation $$ \partial_t w + u \cdot \nabla w = 0.$$

How do you recover $u$ from $w$? and vice versa? Introduce, in 2D, the stream function $u = \nabla^\perp \psi$.

$\psi|_{\partial \Omega} =$ locally constant. We have $\Delta \psi = w$.

Kelvin: A curve $c_0$ at time $t =0$ evolves along the flow to a curve $c_t$ at a later time. We obtain that $\int_{\Gamma_i} \frac{\partial \psi}{\partial N} = \gamma_i$. $\Delta \psi = w$ and some boundary conditions….

I can rewrite the vorticity equation as $\partial_t w + {\psi, w } = 0$ where the bracket is defined as the area of the parelellogram determined by $\nabla w$ and $\nabla \psi$.

Transport interpretation.

Coadjoint orbit.

Theorem (Choffrut-Sverak; GAFA 2012): Let $\overline{w}$ be a “non-degenerate” steady state. Then there exists a $C^\infty$-nhbhd $W$ of $\overline{w}$ such that any coadjoint orbit intersecting $W$ contains exactly one steady state in $W$.

The proof is by an inverse function theorem. I need to say how I am going to implement this function theorem. How do I describe my orbits? How do I describe my steady states? I want to show these are in one-to-one correspondence.

….lots of discussion…..smooth dependence….lots of chatter…..speaker needs to be able to describe more.

Characterization of steady states. The transport equation tells me that there is no dependence of $w$ on $t$ so that $w (\eta_t (x)) = w(x)$, $w = F(\psi)$ provided $\nabla \psi \neq 0$. I impose that $\nabla \psi \neq 0$ in $\Omega$ and I assume that $\Omega$ is an annular domain (with exactly one hole). Steady states are exactly parametrized by $F$.

Characterization of coadjoint orbits.

Transport $\implies A(0) = | [x \cdot w (x) < c ]| = |[x: w \circ \zeta (x) < c]|$

Correspondence.

Steady states ……… F
Orbits …………. A

Steady states can be characterized as conditional critical points of $$ E(w) = \frac{1}{2} \int_\Omega | \nabla \psi |^2 $$ (restricted to an orbit.)

Recall $\Delta \phi = w$. Calculating the first variation leads us to $\delta E = \int \nabla \psi \cdot \nabla \phi = - \int \psi { \alpha, w} = 0 $ (using a permutation property.) Since this is true for all $\alpha$, we find that ${\psi, w} = 0$. Similar ideas come up in the Arnold stability theorem.

Euler as a geodesic configuration space. Lagrangian least action principle. Marchioro-Pulvarenti, Chemin.

Clairaut, Noether, Lie group with (left) invariant metric.

That was the Lagrangian formulation on the tangent space. The real action takes place in the cotangent bundle where we have a Hamiltonian formalism.

Thomas Kappeler: Symplectic techniques for integrable PDEs

(pdf slides)

Aim: Survey of recent results on integraple PDEs obtained by symplectic techniques. The model equation is the $NLS_3^{\pm}(T)$.

Topics:

Phase portrait/ space of orbits; construction of normal coordinates
Asymptotic properties of solutions
KAM theorem

NLS as a Hamiltonian system.

Phase space: $L^2_C$. Pairs of functions. More generally, a function space.

There is a canonical Poisson bracket

$$ {F,G}(\phi) = -i \int_0^1 \partial_1 F \partial_2 G - \partial_2 F \partial_1 G) \phi dx. $$

Defocusing NLS. Focusing NLS.

Defocusing NLS as an integrable PDE on T. Grebert-Kappeler-Poschel “The defocusing NLS equation and its normal form” to appear in EMS.

Focusing NLS as integrable PDE: only few results.

Part 1. Review of NF for defocusing NLS.

Theorem (GKP): There exists a canonical map (closely related to the Fourier transform) which reveals that defocusing NLS may be viewed as a system of infintely many coupled oscillators.

Steps of proof:

Local part. We have to construct these coordinates $x_n (\phi)$ and $y_n (\phi)$. How to build these coordinates?

Global part. We have a global chart.

Zakharov-Shabat operator (ZS)

Lax pair for NLS
Periodic spectrum of $L(\phi)$ on [0,2].

Counting Lemma: He describes the spectrum with a picture on the board. Floquet theory.

characteristic function
two-sheeted spectral curve

von Neumann-Wigner 1929

The eigenvalues come in pairs. Asymptotically, they are like $n\pi + l_n^2$.

Construction of actions/angles

Choose cycles $a_n$
Cycles induce (i) actions and (ii) 1-forms.
1-forms induce angles.

Birkhoff coordinates

Euclidean versions of these action angle coordinates.

Important features of construction

same cycles $a_n$ were used to define $I_n$ and the 1-forms $\beta_n$.
Cycles/1-forms are defined on the spectral curve and not on phase space.

Part 2. Normal Form for fNLS

There do not exist global Birkhoff coordinates.
The associated Zakharov-Shabat operator $L$ is not necessarily self-adjoint.
Symmetries of $spec_p (L(\phi))$.
$spec_p (L(\phi))$ can be described.

We believe that one can characterize the (local) existence of Birkhoff coordinates near $\phi \iff$ spectral properties of the Zakharov-Shabat operator.

Standard Potentials

Describes how to carry out the construction. The focusing case requires local constructions.

The discussion here is very precise and I don’t think I can convey more than is available on the slides.

…

Part 4. KAM for defocusing NLS

Defocusing NLS is integrable on all of $L^2$.
Question: KAM on $L^2$, not only near equilibrium point 0?
Defocusing NLS frequencies can be expressed using the Birkhoff coordinates.

Proposition: Can check the Kolmogorov and Melnikov conditions and then conclude by analyticity.

Near resonances

$\omega_j - \omega_{-j} = O(1)$ for $ j \rightarrow \pm \infty$ jeopardizes measure estimate of standard KAM theorem for integrable PDE.
Ways to overcome difficulties:
Craig-Wayne-Bourgain method. Bourgain, IMRN 95
Kuksin-Poschel, Berti
Restrict the perturbations Geng-You CMP 06, JDE 05

Kappeler-Liang JDE 12

IAS Workshop on Symplectic Dynamics 2: Monday

Tue, 13 Mar 2012 18:39:20 +0000

I’m participating in a workshop at the Institute for Advanced Study on Symplectic Dynamics. This is part of the special concentration this year organized by Helmut Hofer. I’ll try to write notes on the talks I hear. Apologies to the speakers for typos and misquotations…. Comments (pending approval) are open for suggestions and edits.

IAS School of Mathematics

Workshop web page

10:15 - 11:15 Peter Constantin, Princeton University, “Long time, vanishing viscosity limits” abstract
11:30 - 12:30 Steve Preston, University of Colorado, “The inextensible string as a toy model of fluids” abstract
2:30 - 3:30 Emanuele Caglioti, University of Rome, “Long time behavior of solutions of Vlasov-like equations” abstract
4:30 - 5:30 Roberto Camassa, University of North Carolina, “Large amplitude internal waves and their stability” abstract

Peter Constantin: Long time, vanishing viscosity limits

(pdf slides)

Navier-Stokes

Navier-Stokes equations. $\nu$ multiplies the damping term. Studying it in $R^d$ or $T^d$, $d=2,3$. We are interested in $T \rightarrow \infty$ and $\nu \rightarrow 0$. (Reynolds number $\frac{UL}{\nu}$.)

Limits: selected stationary statistical solutions.

$$\lim_{Re \rightarrow \infty} \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T \phi( S^{NS} (t) ) dt.$$

Anomalous dissipation of energy:

$$ \lim_{\nu \rightarrow 0} \lim_{T \rightarrow \infty} \frac{\nu}{T} \int_0^T \int_{R^3} |\nabla u (x,t)|^2 dx dt = \epsilon > 0.$$

(K41)

2d Navier Stokes

No anomalous dissipation of energy. Enstrophy balance. Anomalous dissipation of enstrophy? Kraichnan 68: Yes. Bernard 00: add (extra…to be described…discussion….friction from the bottom) damping and then no.

Constantin-Ramos 07: Bernard was right. Heuristics.

We are talking about weak solutions. 2 reasons to talk about these. The singularities really describe the phenomena, e.g. shocks in hyperbolic conservation laws. Deduce from the few conservation laws that solutions exist and they are nice and smooth but we can’t so we content ourselves with what we can build.

Active Scalars

$$\partial_t \theta + u \cdot \nabla \theta = 0$$

Examples: 2d Euler, SQG.

$$ u = \Lambda^\gamma R^\perp \theta. $$

(Here $R$ is the Riesz transform, $\Lambda = \sqrt{-\Delta}$.)

Weak solutions for SQG. Resnick 95. Almost Lipschitz on $L^2$. Euler and NS does not have this weak continuity property. We don’t have uniqueness. If you have unique weak solutions, then you are golden. I’m going to talk about statistical solutions eventually.

Theorem (Chae-C-Cordoba-Gancedo-Wu): Let $\theta_0 \in L^2 (T^2)$. There exists a global $L^2$-weak solutions of the generalized SQG.

….too fast to type. Generalizes result of Resnick. Remarks…fast. Friedlander-Vicol. Moffat.

A commutator estimate

Summarizes the ideas of the proof. Stream function representation. Cancellations….end up with a commutator….end up with a weak vs. strong.

Damped Driven Navier-Stokes Equation

Standard Navier-Stokes with forcing and an added term $+ \gamma u$, which represents the damping.

You get some a priori bounds which are independent of the viscosity by following the usual arguments.

Stationary Solutions (a warm up)

$$ \gamma \omega + u \cdot \nabla \omega - \nu \Delta \omega - g =0.$$

Absence of anomalous dissipation

Theorem: $$ \lim_{\nu \rightarrow 0} \nu \int_{R^2} |\nabla \omega^\nu |^2 dx = 0.$$

The key idea is to extract extra structure by taking limits.

Statistical Stationary Solutions

Definition: A stational statisitcal solution (SSS) of the damped, driven NS equation on the phase space of vorticity is a probability meausre $\mu^\nu$ on $L^2 (R^d$ such that )…..some conditions indicating that the equation holds in a weak sense against “cylindrical test functions”.

Ideas of proof….pretty fast….hard to keep up while typing. He emphasizes the role of a nonlinear function $\beta$ which he uses to map from “$L^2$ to $L^\infty$”

One Key idea. Use $J(u \omega) - J(u) J(\omega)$ when you are considering a mollifier $J$. This is a useul trick. “Quadratic flux formula”

The hard part of the proof is to prove the enstrophy balance.

The delicate dance of the $\beta$ parameter reminds me of the role of the smoothing operator $I_N$ in the $I$-method. I’d like to understand this better….

The punch line is that you only need this “old language” for the proof. At the end, we obtain a proof of the absence of anomalous dissipation. I’d like to prove corresponding results for quasigeostrophic.

Remarks:

The existence of weak solutions for damped, driven Euler equations: Barcilon-Constantin-Titi
For absence of anomalous dissipation: Renormalized weak solutions. Enstrophy balance.
Results for gSQG. Mentions Caffarelli-Vasseur, Nazorov-Volberg-? on this equation.

Questions/Discussion

A suggested direction: Stability implies regularity. If you assume that NS solutions are $L^2$ stable then you should be able to prove regularity. Conjecture: Fix viscosity. Imagine you have a time horizon and you have a constant. Lipschitz regularity of the flow map implies regularity.

Steve Preston: The inextensible string as a toy model of fluids

(pdf slides talk; joint work with Ralph Saxton)

The original idea that the inextensiblestring might have soemthing to do with fluids was due to V. Yudovich. This problem was introduced to the speaker by A. Shnirelman. The part of fluids I want to have a toy model for is the geometric viewpoint.

Geometric aspects of fluid mechanics

At a point in a fluid, you have a velocity felid and we minagine this vector pushes a fluid element along that direction. You put an $L^2$-Riemannian metric on the space of maps $C^\infty (M, M)$ and you define geodesics wrt this metric.

Volumorphisms. He views this as a “submanifold” in the space of all maps. A lot of things break down here but formally we think of this curve of volume preserving maps inside the larger flat space of all maps. Hodge decomposition allows us to decompose an arbitrary map into one along the volumorphism submanifold and another involving a divergence.

Misiolek-Himonas

Ebin-Marsden

Riemannian exponential map.

Eulerian viewpoint is easy to formulate, but the Lagrangian form is more convenient for geometry. He emphasized the loss of dierivatives.

“Smoothness of the exponential map is the most basic requirement for doing infinite-dimensional Riemannian geometry rigorously.”

There is a gap between the topology and geometry of volumorphisms.

Topology. We want the volumorphisms to be a smooth submanifold of the space of smooth maps from $M$ to $M$. We can make this rigorous if we enlarge to Sobolev $H^s$ diffeomorphisms with $s > \frac{1}{2} {\mbox{dim}}(M) + 1$ to ensure that $\eta \in C^1$. For smaller $s$, we don’t get a smooth submanifold.
Geometry. The metric is defined only in terms of $L^2$ distance. So geometrycially, we should consider measureable maps $\eta: M \rightarrow M$ which preserve the measure. These may not even be bijections so that manifold structure fails (not all tangent spaces are isomorphic).

If $M$ is three-dimensional, the $L^2$ closure of smooth volumorhpisms is the space of all measure-preserving measureable maps (Shnirelman). Fluids should not behave like that! Or maybe they could come close?

In 2d, we don’t understand this so well.

Lagrangian Averaged Euler equations

We introduce a parameter $\alpha$ and define an alternative Riemannian metric which has an $H^1$ inner product (time $\alpha$) and we consider this only on the Volumorphism “submanifold”. The associated geodesic equation leads to the LAE-$\alpha$. “The idea is to average over small scales of a fluid; as $\alpha \rightarrow 0$ we expect the solutions to approach the usual Euler equation solutions.” When these were introduced, it was expected that 3d global existence would be easier than for Euler but this has not turned out to be the case.

To better understand what is going on, we want simpler one dimensional examples.

Camassa-Holm: $ u_t - u_{txx} + 3 u u_x - 2 u_x u_{xx} - u u_{xxx} = 0$
Constantin-Lax-Majda: $\omega_t = \omega H \omega, ~ u_x = H \omega$$

In the CLM equation $H$ is the Hilbert transform. Modified: $\omega_t - \frac{1}{2} u \omega_x = \omega u_x, ~ u_x = H \omega$. This is a geodesic equation on the space of diffeomorphisms on $S^1$ with a right-invariant $H^{1/2}$ metric.

….quick transition to the inextensible curves…..moving faster….

Unit speed parametrization. This constraint forces the curve the generate its own tension to satisfy the constraint. This is similar to the pressure which results from the zero divergence condition.

Geodesic equation for $L^2$ string

Orthogonal acceleration $\implies$

$$ \eta_{tt} = \partial_x (\sigma \eta_x)$$

where $$ \sigma_{xx} - |\eta_{xx}|^2 \sigma = - |\eta_{xt}|^2. $$ “Here the tension $\sigma$ is analogous to the pressure in the Euler equation, determined nonlocally by a purely spatial differential equation.

WE can think of this as an approximation of the nonlinear wave quation, in the same way as the incompressible Euler equation is an approximation of the compressible Euler equation.”

(This is an interesting nonlocal equation so it should be mentioned on the developing nonlocal equations wiki.)

Finite-d approximate model through a system of rigid rods oscillating like coupled pendula.

In $R^2$, we can write $\eta_x = (\cos \theta, \sin \theta)$ (which enforces the constraint) and rewrite the equations. Some discussion of boundary conditions. The boundary conditions then determine nonlinear constraints on the parameter $\theta$.

Motion of whips and chains. Preston JDE 2011
Bootstrap Chao Ma, PhD thesis forthcoming.

Movie:

$L^2$ whip. The whip cracks. The curvature becomes singular. Thess-Zikanov-Nepomnyashchy The loops make the crack!

So, I’d like to revisit the Lagrangian averaging in the setting of the inextensible string. He again adds the $H^1$ inner product to the Riemannain metric on the space of curves. This produces a new geodesic equation, which does not resemble the earlier wave equation. It does still have the nonlocal $\sigma$,…technical slide….

Preston-Saxton, DCDS-A (to appear) Some GWP results.

Movie:

Same initial condition and the loops don’t get pinched.

Emanuele Caglioti: Long time behavior of solutions of Vlasov-like equations

(pdf slide talk)

Outline

one slide, overview of the talk.

Vlasov-Poisson and 3d Euler

The Vlasov equation $$\partial_t f + v \partial_x f + F_f \partial_v f = 0$$

Here $f(x,v): S^1 \times R$ is the phase space density: $$ \rho (x) = \int dv f(x,v) $$ is the space density $$ F_f (x) = \int dy \rho(y) \mathcal{F} (x-y) $$ is the force.

Vlasov-Poisson Equation (VPE)

$\mathcal{F} = \partial_x \mathcal{V}, ~ \mathcal{V} = \partial_{xx}^{-1} \delta$. For $x \in [0, 2 \pi)$….ack slide change….

2d Euler equation is another example. The vorticity $\omega$ is transported along the flow.

The density $f(x,v,t)$ is transported along the trajectories of an Hamiltonian system: $$ \dot{x} = v, \dot{v} = F_f (x). $$

The hamiltonian is a functional of the density $f$ itself: self-consistent force field. Therefore the area of the level sets of $f$ is conserved. Same for 2D Euler the vorticity is tranported along the trajectories of an Hamiltonian system.

Mather: What does that mean “self-consistent force field”? Answer: Some discussion about the mean field limit and statistical mechanics. Here is a link to related discussion.

Possible behaviors

Stable stationary solutions of VPE.

Marchioro-Pulvirenti (1986)

$f = g(v) $ is an example.

BGK waves

Bernstein, Greene, Kruskal (1957)

$ f = f_0 (x - u_0 t, v - u_0).$

These solutions satisfy some conditions related to the Hamiltonian.

I try to make a parallel with 2D Euler.

Any radial vorticity: $ \omega = g(\rho), ~ \rho = \sqrt{x^2 + y^2}$ is a stationa soution of 2D Euler.

Kirchoff (1876) showed that elliptiacl patches are rotating solutions of 2D Euler. The patch is stable if $a < 3b$ (parameters refer to the geometry of the ellipse.)

Landau Damping

Landau, on the basis of the analysis of the VPE linearized around an equilibrium conjectured that for initial data close to equilibrium $$ f_0 = f(v) + \epsilon g(x,v) $$ asymptotically the electric field will vanish and the phase space density will become homogeneous. The linear case has ben fully characterized (Maslov- and Fedoryuk.)

Existence of a class of damped analytic solutions has been proved with a scattering approach by Caglioti-Maffei (1998). Hwuang-Velasquez (2009) extended this result to situations close to equilibrium solutions. Initial data cannot be characterized.

Mouhot-Villani (2009) proved that close to equilibrium initil data are exponentially damped (Landau Damping). The result is proved in an analytic framework (also Gevray type regularity).

Lin-Zeng (Recently) have shown that BGK exists for small regularity: $W^{s,p}, ~ s< 1 + \frac{1}{p}$. Therefore, there is NO LANDAU DAMPING for small regularity. This is interesting to me….a long time behavior which is dependent upon regularity.

Matthaeus (1991) simulation.

….Constantin: this is probably hyperviscosity. ok, but the point he wants to convey is not dependent on this issue. Shnirelman: this is a very robust picture. When you simulate NS on 2D torus, it always emerges that there are two vortices like this.

Possible limiting behaviors.

Stationary (stable) solution for VPE and for 2D Euler
time periodic solutions for BGK and Kirkhoff Ellipses for 2D Euler
Landau damping
Is it possible to prove damping to BGK solutions? Many people are working on this.

(This theory seems to be quite analogous to the state of the art in nonlinear Schrodinger and wave equations.)

What can we say in general?

Shnirelman ICM 2010 Mixing operators in $L^2$. Shnirelman’s construction. Bistochastic operators.

Partial ordering, minimal flows.

It is possible to prove that minimal flows are stationary stable solutions of 2D Eler.

A first conjecture: The set of minimal flows is an attractor for the 2D Euler flow (essentially Landau damping conjecture)

Motivation: If the fluid does not go to stationary solutions level lines of vorticity are stretched and stretched and therefore the solution reaches a minimal element.

The conjecture is probably wrong because more complicated behaviors are expected from simulations.

Brenier: True with probability 1. Caglioti: probably not.

Generalized minimal flows (Shnirelman):

The Navier-Stokes equation with random forcing in the null viscosity limit has an attractor which is concentrated on generalized minimal flows. We reformulate this conjecture in the language of Landau Damping from Vlasov.

A strictly related conjecture.

Given $f_0$, let us define $$ \Omega (f_0) = [ \mbox{weak limit points of} f(x,v,t): t \rightarrow + \infty ] $$

Then it is reasonable that generically:

$S(g) \geq S(f_0)$
If $g_1$ and $g_2$….ack slide changed.

Construction of periodic solutions for the HMF model

Morita-Kaneco PRL (2006)

See also Antoniazzi-Fanelli-Barre-Chavanis-Dauxois-Ruffo, PRE (2007)

work in progress with D. Benedetto and P. Butta.

Conclusions:

We might reasonably expect that asymptitocally the dynamics will become a simple motion. It might even be chaotic but with a few degrees of freedom involved.

Roberto Camassa: Large amplitude internal waves and their stability

Article about Carolina Wave tank

(Collaboration with A. Almgren, S. Chen, R. Tiron, C. Viotti.)

Somewhat soft…..filled with movies…..suitable for the end of the day.

Outline

Motivation: practical (quantitative) vs. “paradigm” (qualitative) significance of simple models of wave motion in fluids?

Three examples:

Two layer Euler vs. strongly nonlinear models
Wave induced instabilities
Integral equations:

Introduction. Fluids lab experiments. Cool movie.

Stratified incompressible Euler equations. 2 layer model. As you saw in the movie, diffusion can be ignored on the time scale of the movie.

Grue et. al JFM 1999 (Norway experimental group)

Stanton-Ostrovsky 1998 (Oregon Coast), 150m

ASIAEX 2004 (340m)

Helfrisch-Melville 2006

These papers reveal that large amplitude internal waves exist.

Miyata 1998, Choi-Camassa JFM 1999

Shallow water strongly nonlinear models.

Pictures….pictures….movies.

Richardson Number.

Taylor-Goldstein Eigenvalue problem controls stability.

Edinburgh Arnold Memorial Workshop Notes

Mon, 03 Oct 2011 21:00:10 +0000

I am at an interesting workshop in Edinburgh entitled Dynamical systems and classical mechanics: a conference in celebration of Vladimir Arnold 1937 - 2010. Boris Khesin and Serge Tabachnikov have coordinated a Tribute to Vladimir Arnold which will soon appear in consecutive issues of the Notices of the AMS. These tributes were shared at the workshop and are also available here:

I will post below my notes from the talks. I apologize, especially to the speakers and readers (if any), for errors and typos.

Welcome

Remarks at introduction of workshop by S. Kuksin:

Kuksin highlights the openness of Arnold to discussions with students. A main message from Arnold “Mathematics must be interesting.”

Small program changes: Lai-Sang Yang speaks on Thursday at 1630. Laurent Stolovich speaks on Friday at 930.

Wednesday will have some short afternoon talks. Young people can either speak to me or Hakan Eliasson. We will make a page with a list of the talks.

Program

Monday 03 October

Alexander Shnirelman, Concordia University,Some problems of the fluid mechanics

I am grateful and touched because of the invitation to speak at this conference. I had the influence of Arnold for several decades. I remember that the pace of sdiscoveries in his seminar was so fast. It reminds me a bit of the situation like 500 years ago. There was not enough time to examine in detail the discoveries as they developed. It is a bit similar to the discovery of continents.

1. Arnold’s presntation of basics of the fluid dynamics.

A lagrangian system whose configuariton space is a Lie group $G$ with unit element $e$. Take the kinetic energy defined by a right-invariant Riemanniain metric via $$ E(u) = \frac{1}{2} \langle u , u \rangle = \frac{1}{2} (Au, u)$$ where $A: H \rightarrow H^*$ is the inertia operator. ….oh my this is too fast to follow while typing.

This is a survey of the Arnold approach to fluids. The critical points of the energy on each orbit are steady solutions.

Example: $SO(3)$. Surfaces $S$ are ellipsoids. levels lines. stable points.

We collapse to $n=2$. Take the quadratic form to be the $L^2$ norm. Discussing 2d Euler equation. Arnold called the surfaces isorotated velocity fields. There exists a unique stream function.

The second variation of $E$ on $S$ at the point $u$ is given by the quadratic form $$ \delta^2 E (\phi) = \int_M (\nabla \phi)^2 + \frac{\nabla \psi}{(\nabla \Delta \psi)} (\Delta \phi)^2 dx. $$ The solution $u$ is called Arnold Stable if this form is either poisitive or negative definite. What is $\phi$ here? It is a perturbation of $\psi$ and he writes on the board $\delta \psi = [ \psi, \phi ]$ (but with curly brackets).

2. Difficulties of the Arnold’s Approach

The group $D$ is infinite-dimensional; its topology is usually stronger than topology defined by the Reimannian metric. Hence the existence and uniqueness of geodesics are not certain.
The surfaces $S$ may be nonsmooth, and the partition of $H$ into these surfaces may be locally nontrivial.
It is unclear whether the energy functional $E$ attains a maximum or a minimum on a given orbit.

3. Group as a Banach Manifold

Theorem (Lichtenstein, Giunter, …):

For any initial velocity $u_0 \in X$, where $X$ is one of the above spaces (e.g. Holder, Sobolev, …), there exists $T>0$ and a unqique solution $u(x,t) \in X$ of the Euler equations with initial velocity $u_0$ defined for $|t| < T$.
If $n=2, T = \infty$. (Volibner, Yudovic, Kato, ….)

Mentions Ebin-Marsden.

4. Mixing operators.

If a 2d domain $M$, the vorticity $\omega$ is transported by the flow; it is distorted and effectively missed. He considers a class of operators on $L^2$ given as integral kernel operators with positive kernel (so positive measures) and with marginals which are equal to 1.

He defines a partial order in $L^2$: $f \ll g$ if $f = Kg$ for some $K \in {\bf{K}}$. Now we write $u \ll v$ for two vector fields $ u, v \in V$ if $ curl u \ll curl v$.

By Zorn’s lemman, there exist a minimal flow wrt this ordering.

Theorem Minimal flows are Arnold stable.

Minimal flow is called energy excessive if $F’ \leq 0$; energy deficient if $F’ \geq 0$.

5. Long-time behavior of the flow; mixing of vorticity

Natural conjecture: minimal flows form an attractor. However, this is wrong.

Movie.

Shows a 2d torus. Vorticity on square patches. After some transition period, the solution becomes more or less periodic plus a constant velocity drift. In this experiment, the flow converges to time periodic flow whih is not a stable configuration. More detailed experiments show that the final flow is quasiperiodic with more details. Even possible to find almost periodic. These are not stable flows. There exists a wider class of flows which are attracting.

6. The evidence of irreversibility: Liapunov function.

Defines a Liapunov function. Existence of Liapunov function always shows that there is some kind of irreversibililty. Shows an elemtary example. For a free particle, we can consider $L(x, \dot{x}) = x \cdot \dot{x}$. (This is reminiscent of the Morawetz estimate.)

The first Liapunov functional for the fluid was found by V. Yudovic (1973). The LF is given by $L = \omega \omega_x \omega_y$. It turns out this combination satisfies that its time derivative is given by a square $(\omega^2 \omega_x^2)|_\pi \geq 0$.

7. Generalized minimal flow

Suppose $u_0 \longmapsto u$. Let’s close the orbit of the evolution in $L^2$.

Definition: A flow $u(t)$ with initial veloicty field $u_0$ is called a generalized minimal flow (GMF) if ….curl condition.

slide changes fast.

8. Construction of GMF by pseudoevolution

A process is decribed which produces GMF’s from a given seed data using the curl ordering.

Conjectures:

The set $N$ is an attractor for the Euler equations in the ordinary sense.
GMG are either statiornay or qaiperiodic with at most countable set of periouds.
Stircktly speaking, we have not proven that the set $N$ is a a proper subset of $V$. e.g. there might exist flows which are not GMF. (This is analogous to the Landau damoping recently proved for the Vlasov-Poisson equation by Villani.)

9. Local regularity of partition into isovortical surfaces

The equation was addressed recently by V. Sverak and A. Choffrut (2010). Consider the distribution function for the vorticity $$ \lambda (s) = {\mbox{mes}} [ x \in M : \omega (x) \leq s]. $$

Theorem (Sverak-Choffrut): Steady solutions close to a fixed Arnold stable one are in a smooth 1-1 correspondence with distribution functions $\lambda (s)$.

The proof is difficult and based on Nash-Schwarz implicit function theorem.

10. The structure of the exponential map.

Ebin-Marsden 1970
Ebin-Misiolek-Preston 2008
Misiolek 1993

John Mather, Princeton University,Near a double resonance

(blackboard talk)

I feel honored to speak here, especially since something I have been working on for a long time is called Arnold Diffusion. Some people speak about the problem. What is interesting to me is the program. Arnold posed many problems. There are two aspects that should be highlighted.

They are very interesting.
There is a chance that we could do them.

Generic quasiergodicity of Hamiltonian systems….a famous problem attributed to Boltzmann. This problem seems incredibly hard and seems to be inaccessible. In contrast, there appears to be hope for Arnold diffusion.

I apologize to those of you have heard this talk before. I want to speak about something that I announced in 2003 in Russian and it appeared in English in 2004. I announced some results at that time but, in the meantime I found some mistakes in the proof. I want to speak about corrections to those proofs.

This is a question about small perturbations of integrable systems. Usually, people discuss this in the setting of the Hamiltonian form. I prefer to approach it using the Lagrangian (equivalent form) since my method of approach is variational.

Lagrangian: $$L(\theta, \dot{\theta}, t) = l_0 (\dot{\theta}) + \epsilon P(\theta, \dot{\theta}, t)$$

We consider here the case where $P(\theta, \dot{\theta}, t+1 ) = P(\theta, \dot{\theta}, t)$.

$\theta \in T^n$
$\dot{\theta} \in B^n \subset R^n $

We are looking for solutions of the Euler-Lagrange equation $$ \frac{d}{dt} ( L_{\dot{\theta}} ) = L_\theta. $$ Let’s assume that $d^2 l_0 >0$. (This is a strong restriction; we’d rather like to do it under the assumption that the determinant of the Hessian is nonzero. A great deal of the theory is developed under that assumption. For the methods I use, I need this stronger condition.)

The goal is to somehow show that the solutions go everywhere. That is too strong, but I can prove something along those lines in a special case.

The Lagrangian I am looking at is a small perturbation of an integrable system. For the integrable system, the E-L equation is $$ \frac{d}{dt} ( L_{\dot{\theta}} ) = 0. $$

Arnold Question: Are the orbits confined or do some go everywhere? (This is a vague question; certainly Arnold was more precise.)

In the case when $n=1$, the orbits are confined. This is a consequence of KAM theory. The KAM tori persist. There are Birkhoff regions of instability and the orbits are confined by the KAM tori.

In the case $n>1$, the expectation is that the orbits are not confined. There are results along these lines which show, in some cases, that this is the case. For the program we have in mind, we want to build methods which show this phenomena is somehow generic.

In the case $n=2$ (this is what I had announced in 2003): Generically (assuming the positive Hessian part), the orbits are not confined.

The methods that I use are variational. I want to say a little bit about those tools.

Consider $U_1, U_2, \dots, U_k$ open non-void sets in $B^2$. The construction guides the orbit to only be allowed to move in certain ways. The trick is to choose the conditions in such a ways so that when you minimize within those conditions, the solution stays inside the open set and doesn’t get pushed off to the boundary. This is a method that I introduced in the past in studying twist maps. The conditions are really complicated so you need a guide to tell you what the conditions are. The basic guide involves something called Aubry sets. These are sets contained inside $T^2_\theta \times T_t$. The phase space consists of $T^2 \times B^2 \times T$ and the state space is $T^2 \times T$. The Aubry sets are defined by a global minimizing condition. What shall I say about them? First of all, it is useful to consider invariant probability measures for the Lagrangian system. Let $\mu$ be a probability measure on $T^2 \times B^2 \times T$. You can then define a cohomoology class $c \in H^1 (T^2)$. This allows you to define an average action: $$ A_c (\mu) = \int_{T^2} L d\mu - c $$ …..I don’t ususally write it this way….scratches it out and writes instead $$ A_c (\mu) = \int_{T^2 \times B^2 \times T} (L (\theta, \dot{\theta}, t) - c \dot{\theta}) d\mu (\theta , \dot{\theta}, t). $$ We can then define $$M_c = [~{\mbox{invariant probability measures that minimize}}~ A_c].$$ The support of $M_C \subset T^2 \times B^2 \times T$ and the map turns out to be injective.

Aubry set….defined by a minimizing condition.

Theorem: $M_c \subset Au_c$.

The conditions I build for the minimization process. First you have to know something about the Aubry sets, you can then state what the conditions are. The process was carried out successfully in the case of twist maps. I was able to prove that there could be wandering in the Birkhoff zones of instability. This procedure was also used by Cheng-Yan in the case of a priori unstable systems. They were able to prove a version of Arnold diffusion in these systems. One considers a rotator and a pendulum and take the product of the two. The diffusion is related to the unstable fixed point.

…trouble typing…..discussion of double resonance, with strong so that the resonant linear combination arises with control on the integer prefactors by a constant.

Massimiliano Berti, University of Naples Federico II,Quasi periodic solutions of Hamiltonian PDEs

(Similar to what I saw in France at Ile de Berder, so I will watch rather than type…)

There was some discussion afterwards between me, Massimiliano and Walter Craig. I suggested that Massimiliano’s improvement of Walter’s pseudodifferential result might be reconsidered in the setting of the $MMT_{\alpha, \beta}$ models introduced by Majda-McLaughlin-Tabak. These models are slightly more general and consider parametrized $\alpha$-power dispersion relation with a $\beta$-smoothing operator inside the cubic nonlinearity. This might provide a generalized framework for investigating the relationship between dispersive smoothing and the derivative properties appearing in the nonlinearity. Admittedly, these are not directly physical models but the mathematical motivations for their study seem to keep appearing….

Andrei Agrachev, SISSA Trieste, The long-time behaviour of dissipative systems

A natural mechanical system on a Riemannian manifold. Traectories are curves on this manifold. The kinetic energy is the usual Riemannian length. The potential energy is a function on the manifold.

Hamiltonian $ = \frac{1}{2}|p|^2 + V(q)$ where $p \in T_q^* M$ and $|p| = \max [ \langle p, \xi \rangle: \xi \in T_q M, |\xi | =1 ]$

WE consider this system but with an isotropic dissipation: $$ \dot{p} = - H_q - \alpha p, ~\alpha > 0$$ $$ \dot{q} = H_p. $$

Toy example: $M=R, V(q) = b q$. All solutions converge to one particular solution. Eventually, the particle moves with a fixed velocity. If we perturb the V a little bit, the phase portrait will be very similar. There will be a limiting profile and the structure will be very similar.

Toy example: Pendulum. $V(q) = b \cos q, ~ \frac{\alpha^2}{4} < |b|.$ Then, we don’t have limiting behavior like that beffore. We have instead a vortex. The dissipation brings the trajectory down to the minimum of the potential energy. However, when we have $ \frac{\alpha^2}{4} > |b|,$ we have a different limiting configuration. We obtain a limiting potential “gradient” flow on the circle.

There is strong dissipation in life. The limitig dynamics of systems we observe, like a ship on the ocean, has a transitional period but eventually there is a balance.

We try to view things using the Eulerian viewpoint.

Definition: Potential stationary flow is a gradient vector field $\nabla u$ where $u \in C^2 (M)$ and $[d_q u: q \in M] \subset T^* M$ is an invariant submanifold of our system.

In particular, $\dot{\gamma}(t) = \nabla_{\gamma(t)} u$ implies that $t \longmapsto (d_{\gamma(t)}u, \gamma(t))$ is a solution.

Definition: The curvature of the Hamiltonian $H$ at $p \in T_q^* M$ is a self-adjoint linear operator from the cotangent bundle to the cotangent bundle defined by the formula $$ R^H_{(p,q)} \xi \cal{R} (\xi, p)p + (\nabla_q^2 V) \xi, ~ \xi \in T_q^* M, $$ where $\nabla$ is the covariant derivative and $\cal{R}$ is the Riemannian curvature.

(This is a natural extension fromt he standard symplectic setting to the dissipative systems. Some interesting discussion…what is the connection….natural…only this kind of isotropic dissipative systems.)

Assume that $M$ is complete, $\cal{R}$ and $\nabla^2 V$ are uniformly bounded. (This follows if $M$ is compact.) Let $\Phi_t$ be the flow on the cotangent bundle. We consider a strip defined by $\Omega_c = [ (p,q) \in T^*M: |p| \leq c]$.

Theorem:

If $R^H_{(p,q)} < \frac{\alpha^2}{4} I, ~ \forall (p,q)$ such that $H(p,q) \leq \max V$, then $\exists$ a potential stationary flow $\nabla u$ such that $$ \Phi_t (\Omega_c) \rightarrow [ d_q u : q \in M] ~{\mbox{as}}~ t \rightarrow + \infty$ $$ with an expoenetntial rate, $\forall c>0$.
$[d_q u: q \in M]$ is a normally stable submanifold of $\Phi_t$.
If $M$ is compact and $R^H_{(p,q)} < \frac{\kappa -1)\alpha^2}{\kappa^2} I$ then $ u \in C^k (M)$.
The map $(H,\alpha) \longmapsto u$ is continuous in the $C^2$-topology.

Smaller dissipation: When dissipation is smaller, we have some hopeful hints. Discussion is moving a bit fast for me to type….Markov process…not an invariant measure but the measures can be propagated…

Interesting discussion about the use of measures in the presence of small dissipation limits as a device to probe the structure of the original Hamiltonian systems.

Slides stop….he still has about 10 minutes. He tries to explain the proof. This discussion is reminiscent of a principal theme I will try to convey in my talk. Infinite dimensional systems might be viewed as the envelope system of limits of finite-d systems. When we study the infinite-d system, one strategy of attack is to identify convenient finite-d systems which limit on the infinite-d system. One possible source of these convenient systems might be through appropriate choices of isotropic dissipative systems, which truncate high frequencies.

Some discussion striving to describe the “curvature” of a Hamiltonian system….family of vertical and horizontal Lagrangian distributions. Curvature of dissipative systems is easily accessed…..cheating a bit, but look in the paper for the details. Levi-Civita connection is tangent to the zero section.

Walter Craig, McMaster University,The water wave problem as a Hamiltonian system

I thought I would speak a bit about Arnold’s influence on my mathematical life. We were given his Mathematical Methods book in graduate school. His perspective has pervaded our approach to problems.

(joint work with Catherine Sulem; along with Alessandro Selvitella and Yun Wang)

Outline:

Two ODEs
Euler’s equations
Zakharov’s Hamiltonian
Partial Differential equations as Hamiltonian systems
Birkhoff Normal Forms
Implications of the normal form
The KdV scaling limit

Two ODEs

$$ \dot{z} = z^2, z(0) = \epsilon $$ versus $$ \dot{w} = w^3, w(0) = \epsilon $$ …..ack slide changed and I missed the point.

Euler’s equations

Newton’s laws, Eulerian coordinates, incompressible fluid. We work on a pre-Columbian model of the earth.

Free surface water waves. We imagine that the velocity field is irrotational (oceanographers do this) so we can recast this as a potential flow. The bottom is not a sponge, so no penetration and we assume that the fluid velocity has no normal component at the bottom.

Free surface conditons: Kinetic BC at the top; Bernoulli condition.

Hamiltonian systems: Zakharov 1968. This was a poorly understood paper which has emerged as being very important.

Goals: explain this fact; use it to understand the PDEs.

Partial Differential equations as Hamiltonian systems

Zakharov’s Hamiltonian

The energy functional $H= K + P$ so it should be $$ H = \int \int_{-h}^{\eta(x)} \frac{1}{2} |\nabla \phi |^2 dy dx + \int_x \frac{g}{2} \eta^2 dx. $$

This is pretty clear but the difficulty is what are the choices of variables?

Zakharov’s choice: $$ z = (\eta(x), \xi(x) = \phi(x, \eta(x))).$$ That is $\phi = \phi[\eta, \xi] (x,y).$

In these coordinates, we can realize the PDE for Euler flow for the free surface as a Hamiltonian system in Darboux coordinates. The subtlety is how to differentiate the Hamiltonian wrt the canonical coordinates.

Other Hamiltonian PDEs:

Boussinesq system; KdV equation; NLS; …

Dirichlet-Neumann oeprator:

Laplace’s equation on the fluid domain $-h < y < \eta(x)$. $$ \xi(x) \longmapsto \phi(x, y) \longmapsto N \cdot \nabla \phi (1 + |\nabla_x \eta |^2)^{1/2} = G(\eta) \xi (x). $$
In Zakharov’s coordinates, we can express the Hamiltonian in terms of the D-N operator $G$ as $$ H(\eta, \xi) = \int \frac{1}{2} \xi G(\eta) \xi \frac{g}{2} \eta^2 dx.$$
The water wave system rewritten: $$ \partial_t \eta = G(\eta) \xi ,$$ $$ \partial_t \xi = -g \eta - {\mbox{grad}}_\eta K. $$ (This discussion is closely related to a variational formula of Hadamard from 1911, 1916)

Lemma (Properties of the Dirichlet-Neumann operator): A singular integral operator $G(\eta)$ related to the Green’s function.

Hermitian symmetric.
$G(\eta) \geq 0$ and $G(\eta) 1 = 0$.
$G(\eta): H^1_\xi \rightarrow L^2_\xi$ is analytic in $\eta$ for $\eta \in C^1$ [using a theorem of Christ-Journé (1987)]: $$ G(\eta) \xi = G^{(0)} \xi + G^{(1)}…$$

ack….slide changed.

Conservation Laws:

Mass is conserved. Mass is $\int \eta dx.$
Momentum is conserved. Momentum is $\int \eta \partial_x \xi dx.$
Energy is conserved.

(Poisson bracket calculations are quite direct.)

Taylor Expansion of the Hamiltonian:

From analyticity, we can expand around the stationary zero solution using the Taylor expansion of $G$.

Flow of the Harmonic oscillator. He is considering here the linearized problem and showing that we can explictly solve this problem using Fourier/superposition methods. We encounter a Fourier series with rotating phases. The typical solution is almost periodic.

Basic facts:

The flow preserves the (linearized) energy.
Actions are preserved. (This is the moment map.) Therefore, all Sobolev norms are preserved.
Phases involve linearly in time.

Basic Questions:

Add in the perturbations. Do any of those orbits persist? This turns out to be quite hard. In fact, it is challenging to show that any of them persist. The progress on these questions have been made using KAM theory. We know that there exist periodic solutions.

Do there exist quasiperiodic or almost periodic solutions?
Given a point $z^0$ in some phase space. Does the flow exist in M? This is hard.
Does it exist globally in time? This is basically open, although there are some recent advances. Do we have stability? Do we have a Nekhoroshev stability property?
Can you make the actions grow? Weak turbulence. Growth of Sobolev norms?

Birkhoff Normal Forms

Fix the dimension to $d=2$, so we have $x \in R$. We restrict to the periodic-in-$x$ case. We want to perform canonical transformations to move the Hamiltonian into a normal form at least in some neighborhood of the origin.

Conditions:

Make the transformation canonical.
Make the new Hamiltonian have the same linearization plus only resonant terms up to some order with a new truncation/residual error.
If $Z^{(3)} = 0$, we will have a chance to get longer existence intervals based on the analogy of the first ODEs at the beginning of the talk.

This transformation process is called the reduction to Birkhoff normal form.

Theorem (Craig-Sulem 2009):

Let $d=2$ (and $h = + \infty$) and fix $r>3/2$. Then, there exists a neighborhood of the ball at the origin in $H^r$ on which we have a Birkhoff normal form which kills off the quadratic nonlinear terms resulting a cubic equation.

Note: This transformation mixes the variables $\eta$ and $\xi$.

Outline of the proof: flying slides…..cohomological equation turns out to be a linear equation. THere are no nonzero $m=3$ resonances. It turns out to be rather challenging to show that the flow of the vector field exists.

Implications of the normal form

Long time existence theorem. Work in progress.

We should be able to build solutions that last for time intervals on the order of $\epsilon^{-2}$. I want this time so that I can study the NLS limit of the water wave equation. On this time scale, we would like to have a nice justification. This justification requires the desired long time existence result.

Wu 2009: Small Sobolev data lasts for exponentially long times.

Germain-Masmoudi-Shatah 2009:

For $d=3$, small Sobolev data exist globally in time.

The difference is that I am on a compact domain. Wu is in a dispersive situation.

Nathan Totz & Sijue Wu have done the NLS limit in the non-periodic case. Schneider-Wayne also have results in this direction.

The KdV scaling limit

pretty fast slide switching….but nice moves. He shows how the KdV Hamiltonian can emerge from the water wave Hamiltonian. Now perform the above sequence of transformations on the Birkhoff normal form for water waves. In the limit as the small parameter goes to zero, the water wave Hamiltonian collapses to the KdV Hamiltonian.

Tuesday 04 October

Yann Brenier, University of Nice,From incompressible fluids to dust

It is a great honor for me to be here. I would like to discuss two issues that were familiar to V.I. Arnold. We saw fluids discussed yesterday in Shnirelman’s talk. Dust is a singularity of the Hamilton-Jacobi equation. (Nepecmponcka perestroika)

First part: euler equations and minimizing geodesics for volume preserving maps

Euler equations of incompressible fluid mechanics
Leas action principles
Geometric analysis issues
Minimizing geodesics: existence and uniqueness results for the pressure gradient

Euler’s Equation: Geometric Definition.

The fluid is moving inside a box denoted by $D$. We consider incompressible motion. This is viewed as a time dependent map $M_t: D \rightarrow D$. Points in $D$ are called $a$. $M_t$ is viewed as a map in the Hilbert space $H = L^2 (D, R^d)$, valued in the subset $VPM(D)$ of all Lebesgue measure-preserving maps.

Solutions of theEuler equatiosn, introduced in 1755, correspond to those curves $t \rightarrow M_t \in VPM (D)$ for which there exists a time dependent scalar function $p_t$ called the “pressure field” defined on D such that $$ \frac{d^2}{dt^2}M_t + (\nabla p_t) \circ M_t =0 $$ where $\nabla$ is the gradient operator on $R^d$ (wrt Euclidean norm).

The Principle of Least Action.

Theorem: Assume $D$ convex. The $(M_t, p_t)$ be asolution of the E equations, with constant $\lambda$ such that $$ \sum \frac{\partial^2 p_t}{\partial_i \partial_j} \xi_i \xi_j \leq \lambda |\xi|^2 $$ Then $M_t$ is the unique minimizer, among all curves along $VPM(D)$ that conincide with $M_t$ at the endpoints $t = t_0, t=t_1$ of the following action $$ \frac{1}{2} \int_{t_0}^{t_1} \int_D | \frac{dM_t (x)}{dt}|^2 dx dt. $$

In other words, such a curve is nothing but a (constant speed) geodesic along $VPM(D)$ wrt metric induced by $H = L^2 (D, R^d)$.

Arnold 1966, Ebin-Marsden 1970, Arnold-Khesin book 1998.

The Dual Action

Minimizing the actrion can be written as a saddle point problem, just by using a time-dependent Lagrange multiplier to relzs $$ \inf_M \sup_p \int_{t_0}^{t_1} \int_D [\frac{1}{2} | \frac{dM_t (x)}{dt}|^2 - p_t (M_t (x)) + p_t (x) ] dx dt. $$ This is trivially bounded fro below by $$ \sup_p \inf_M (same) $$ which naturally leads to a dual least action priciple.

Theorem: Using exactly the same conditions ($D$ convex and $(t_1 - t_0)^2 \lambda < \pi^2$, the pressure $p$ is the unique maximizer of the concave dual action $$ I[p] = \int_D J_p (M_{t_0} (x), M_{t_1} (x)) dx + \int_{t_0}^{t_1} \int_D p_t (x) dx dt, $$ where $$ J_p (y,z) = \inf \int_{t} ( \frac{1}{2} |\frac{d\xi_t}{dt}|^2 - p_t (\xi_t)) dt $$ where the infimum is taken over all curves $\xi_t \in D$ such that $\xi_{t_0} = y \in D, \xi_{t_1} = y \in D$.

The proof is elementary and follows from 1d Poincaré inequality.

Geometric Analysis Issues 1

Density of diffeomorphisms in $VPM(D)$. $SDiff(D)$ is the set of volume preserving orientation preserving diffeomorphisms. This is more refined than the $VPM(D)$ condition. For $d \geq 2$, it turns out that $VPM$ is the $L^2$ closure of $SDiff$. The identification of the closure of $SDiff (D)$ for the a prior finer geoesic distance induced by $L^2$ is a much more difficult issue. For simple (say contractile) domains $D$, this closure is still $VPM(D)$ for $d \geq 3$ (but defintely not for $d=2$) as show by Shnirelman in his land mark paper (Math USSR Sb 1985). These results have striking consequences: in particular maps of form $$ M(x) = (h(x_1), x_2, x_3) $$ where $h$ is any Lebesgue-measure preserving map of the unit interval, are in the closure of $SDiff([0,1]^3)$. (Thus, even though we are interested in volume preserving maps, we have to open our eyes to all Lebesgue measure preserving map of the unit interval. This is a much much richer class than $SDiff$!)
Density of permutations in $VPM(D)$. Another interesting subset of $VPM([0,1]^3)$ is made of all “permutations” of all dyadic divisions of the unit cube in sub-cubes of equal volumes. You divide the buce into dyadic sub-cubes, like a Rubick’s cube. To every permutation, you define a permutation which shuffles the cubes. It turns out that the union of these permutations taken over all scales defines a dense set of maps in $VPM$! He shows some remarkable gifs where dust appears related to the orientation reversal.
Geodesic completeness. Big issue….global well-posedness of E.
Minimizing Geodesics. (Shnirelman Math USSR Sb 1986) The 3d case turns out to be “easy” with a crucial use of the convex structure of the dual problem. The case $d=2$ is clearly linked to symplectic geometry and seems extremely difficult: a fascinating strategy has been developed by Shnirelman, by adding braid constraints to the minimization problem, which certainly deserves further investigations.

Approximate Minimizing Geodesics:

fast slide…The existence of such approximations is in no way trivial and is a consequence of a key density result due to Shnirelman (GAFA 1994).

Main Theorem: Let us assume $D$ to be convex, with $ d \geq 3$, fix $t_0 = 1, ~ t_1 = 1$ and consider maps $M_0, M_1 \in VPM (D)$. Then there is aunique pressure gradient $\nabla p_t$ such that for all $\epsilon$-minimizing geodesics, we have in the sense of distributions $$ \frac{d^2 M_t^\epsilon}{dt^2} \circ (M^\epsilon_t)^{-1} + \nabla p_t \rightarrow 0, \epsilon \rightarrow 0.$$

Yann insists that this has “nothing to do with geodesic completeness”. I am confused…..

Minimizing Geodesics: Final Comments

Uniqueness of the pressure gradient. This is a remarkable feature of the theory. There is no equivalent result for finite-d configuration spaces such as $SO(3)$,on which geodesic curves (for appropriate metrics) correspond to the motion of solid bodies in classical mechanics. WE believe this strange phenomenon to be the consequence of the “hidden convexity” of the problem in dimension 3 and more.
Limited regularity of the pressure gradient.

Some references.

Second Part: From incompressible fluids to dust

Gravitating particles as a natural approximation of Euler incompressible fluids ….fast slides…..jet lag….fascinating….Yann is a fast thinker…

Penalization of the Euler Action

Use a penalty method to try to approximate “geodesics” on discrete sets using permutations.

Monge-Ampere (instead of Poisson) nonlinear correction to the classical Newton gravitation.

Rafael de la Llave, Georgia Institute of Technology,Arnold diffusion in a-priori unstable Hamiltonian systems of high dimension

(joint work with Delshams, de la Llave, T.M. Seara)

(Related collaborators: Elisaget Canalias, Marian Gidea, Gemma Huguet, Vadim Kaloshin, …)

Instability for a priori unstable Hamiltonian systems

We consider a periodic in tim perturbation of $n$ pendula and a $d$-dimensional rotor described by non-autonomous Hamitonian, $$ H(p,q, I, \phi, t , \epsilon) = P(p,q) + h(I) + \epsilon Q (p,q, I, phi, t, \epsilon) $$ with $$P(p,q) = \sum P_j (p_j, q_j), ~ P_j = \pm (\frac{1}{2} p_j^2 + V_j (q_j)). $$

Elemntary and regularity assumptions.

H1: Assume that the functions $h, V_j, Q$ are $C^r$ in their corresponing domains with $ r \geq r_0$ sufficiently large.
H2: Assume that the potentials $V_j$ have non-degenerate local maxima, say at $q_j = 0$, each of which gives rise to a homoclinc orbit of the pendulum $P_j$: They are penduli, they have critical points and have homoclinc connections.
“The enemies to this problem are the KAM and the Nekoroshev. Of course, they are my friends in other talks…”
H3: The mapping $I \rightarrow \omega(I) = …$ ack slide change.
H4: The function $Q$ is assumed to be a trigonometric polynomial. (This can be removed)

Remark: [Delshams-Llave-S06], [Delshams-Huguet09], [Gidea-Llave06].

Melnikov Potential:

Poncare-Arnold-Melnikov. This basically measures the effect of the perturbation on a homoclinic orbit at first order. (Big integral….too long to type this fast.)

H5: Assume that the system of equations $$ \frac{\partial}{\partial \tau} L(\tau, I, \phi, s) = 0 $$ admits a nondegenerate solution. This allows us to eliminate the $\tau$ in terms of the other variables.

Poincaré reduced function

H6:
H7: $Q$ satisfies some nondegeneracy assumptions.
H8: You don’t want the resonances to be flat.

….I can’t really keep up….so I will just watch.

Tenyson 83 Probed many mechanisms of diffusion by observing numerically.

Chirikov 82

Vadim Kaloshin, University of Maryland,Hausdorff dimension of oscillatory motions for three body problems

(blackboard talk)

I have a deep admiration of V.I. Arnold. He was a “god of mathematics and still is.” Here is a list of topics that have occupied my interest for research. They are all explicitly linked with ideas of Arnold.

My undergraduate thesis was on a topic called prevalence, a notion of probability one in infinite dimensional spaces. My project emerged from Arnold’s note that one could understand genericity through this notion.
Hilbert-Arnold Problem. This was my first problem I studied with Ilyashenko. This was motivated by the 2nd part of Hilbert 16th problem. You look at a family of vector fields on $x \in S^2, ~ \epsilon \in B^k$. You consider $\dot{x} = v(x, \epsilon)$. Generic fanily of $C^\infty$ v. fields has $LC(\epsilon) < \infty.$
Growth of the number of periodic points. $M$ is a compact manifold. You look at $f: M \rightarrow M; f \in Diff (M)$. You look at $P_n (f) = {\mbox{Number}} [x: f^n x = x]$. How quickly $P_n(f)$ generically?
Arnold Diffusion.

As you can see, most of my research is either inspired by or directly leads from questions suggested by Arnold as interesting directions.

Qualitative analysis of 3-body problem. Let $q_i \in R^d, ~ d =2,3$ are point masses. Each point has mass $m_i$. The Newton law then gives the dynamical law $$ m_i \frac{d^2}{dt^2} q = - \sum m_i m_j \frac{q_i - q_j}{|q_i - q_j |^3}. $$

Kepler motions:

2 Body problem (2BP). $m_0 q_0 + m_1 q_1 = 1$. $$ H(q, \dot{q}) = \frac{\dot{q}^2}{2} - \frac{1}{|q|} $$ Three cases for the 2BP:

$H<0$ either circular or elliptic.
$H=0$ parabolic; escapes to infinity with zero velocity.
$H<0$ hyperbolic; escapes to infinity with nonzero velocity.

Three Body Problem: (Sun-Jupiter-Comet)

Four types of motion:

B: $\sup_{\pm t>0} |q_i (t)| < K < + \infty$
Parabolic: escapes to infinity with zero velocity at infinity.
Hyperbolic: escapes to infinity with nonzero velocity at infinity.
Oscillatory: $\limsup_{t \rightarrow \infty} |q_i (t)| = \infty; ~ \liminf_{t \rightarrow \infty} |q_i (t)| < + \infty $.

What kind of behavior is possible in the future? What kind of behavior is possible in the past?

A famous result of (missed the names…) showed that there are solutions with any of these four behaviors in either direction of time infinity.

Shows a table. He reports that every one of the boxes (except one) in the table have been shown to have positive measure. He will focus on the case whether the situation with oscillatory motions in the past and in the future. We want to know whether this event has positive measure or not. There was a conjecture of Kolmogorov: He conjectured that this was expected to have measure zero.

Shows two papers by Alexeev. The French version has no attribution to Kolmogorov. The English version has attribution to Kolmogorov. Katok says you should attribute this to Kolmogorov. The English version was published after his death.

(joint work with A. Gonodetski; preprint)

Kolmogorov conjecture: $Mes OS = 0$.

Main Result 1. Often 2 degree of freedom 3 body problem have Hausdorff Dimension maximal possible.

Leading Idea: Build a “fat” Cantor set $\Lambda \ni \infty$ with ergodic dynamics.

If one could produce an ergodic component with positive measure, one could perhaps prove a counterxample to Kolmogorov’s conjecture.

Second version of the main result:

Remark: 2 degrees of freedom Hamiltonian dynamics locally reduces to a 2 dimensional area preserving map. My analysis will concern those maps since they have some advantages, for example I can draw pictures.

Newhouse domains in dissipative setting. Suppose $f:M^2 \rightarrow M^2$. Suppose $f$ has a homoclinic tangency (HT). If $f$ has a saddle point then $f^k p = p$ such that unstalbe and stable manifolds satisfy….. RETURN HERE….Newhouse domains…..

Main Result 2: 2 dof 3BPs have Newhouse domains.

Remark: Duarte proved a conservative Newhouse phenomenon. (20 year interval between Newhouse and Duarte.)

Newhouse domains give rise to striking dynamical examples.

1st Model (Sitnikov): There is a beautiful book by Moser which gives an example of oscillatory motions. You have two bodies $q_0, q_1$ in elliptic orbits with eccentricity $e$. The masses $m_0 = m_1 = 1.$ The third body $q_2 = (0,0,z)$ lies on the $z$ axis. $$ H(t, z, \dot{z}) = \frac{\dot{z}^2}{2} - \frac{1}{\sqrt{z^2 + r_e^2 (t)}}. $$

Theorem 1 (GK): $\exists$ open nonvoid $\cal{N} \subset (0,1)$ such that for a generic $e \in \cal{N}$ we have $HD(OS) = 3$. $\cal{N}$ is a subset of a Newhouse domain.

2nd Model (Restricted planar circular 3BP): $m_2 = 0$ (restricted, comet). $q_0, q_1$ move in circular orbits. Motions are planar. In a rotating frame, these masses are fixed. Set $m_0 + m_1 =1$ and choose $m_0 = \mu, ~ m_1 = 1 - \mu$. The parameter $\mu$ is called the mass ratio. Form the so called Jacobi constant $J(x,y, \dot{x}, \dot{y}) = \frac{\dot{x}^2 + \dot{y}^2}{2} - [\frac{ {x}^2 + {y}^2}{2} + \frac{1-\mu}{d_0} + \frac{\mu}{d_1}]$.

(Here $d_0$ represents the distance from $q_0$ to $q_2$ measured in the rotating frame. $d_1$ relative to $q_1$. )

Theorem 2 (GK): $\exists ~ J_0$ such that $\forall ~ J > J_0$ then $\exists ~ \cal{N}J \subset (0,1)$ with property generic $\mu \in \cal{N}J$ and $HD(OS \cap J(…) = J^*) = 3.$

Meta Theorem. Let $[H_\delta]$ be a 1-parameter family of Hamiltonian systems of 2 dof (or 1.5 dof) satisfy Hypothesis:

H1: As $\delta \rightarrow 0$, the limiting Hamiltonian $H_0$ is integrable with a separatrix loop.
H2: For $\delta \neq 0$, we want the separatrix to split tranversally.
H3: Melnikov function satisfies a certain open condition.

Then for a generic $\delta$ in an open nonempty set, $H_\delta$ has a hyperbolic (nonzero Lyapunov exponent) set of HD = 3.

Ideas from proof (Sitnikov):

$e = 0, ~ H = \frac{\dot{z}^2}{2} - \frac{1}{\sqrt{z^2 + 0.25}}$. He draws a picture in the $(z, \dot{z})$-plane and identifies the region $H<0$ and the region $H>0$ and highlights the “separatrix loop”. He glues the points at infinity at $ z = \pm \infty$ to highlight this as a separatrix loop. Constructing oscillatory motions corresponds to building orbits that come arbitrarily close to this point.

$C^2 - \lambda$ - Lemma…. why do we need this? We need quadratic tangency. This is an explicit system so we can’t use genericity.

Marc Chaperon, Université Paris 7,Generalised Hopf bifurcations

It’s a great honor to be here. I admired Vladimir Arnold very much. We liked each other. What I will speak about appears in the MMF v11(3) in memory of Arnold. The reason I became a mathematician was because of Thom but the reaosn why I persisted was probably because of ARnold. It was amazing how much energy he had. When he came to Paris, he knew more about it than I did, more than most natives. He was some kid of wunderkind and remained so his whole life.

Motivation: interest in the coupling of oscillators.

Chenciner-Iooss 1979

….I’m a bit tired so stopped typing.

Anton Zorich, University of Rennes,Lyapunov exponents of the Hodge bundle

(joint work with Alex Eskin and Maxim Kontsevich)

I am jealous towards my colleagues. I can’t claim this work was motivated by work of Arnold. But I can report that he was constantly interested in this topic. I enormously regret that, now that the story is complete, I can not tell it to Arnold.

Motivations. Consider a billiard in the plane with $Z^2$-periodic rectangular obstacles.

Theorem (Delcroiz, Hubert, Lelivre 2011): For almost all parameters of the problem, the billiard trajectory ecapes to infinity with a rate of $t^{2/3}$.

How can we capture this $2/3$? The obstacles that can appear in this story must involve rectangles.

Exponents like this have appeared in work by Giovanni Forni.

Geometric interpretation of multiplicative ergodic theorem:

Consider a vector bundle endowed with a flat connecton over a manifold $X^n$. Having a flow on the base, we can take a fiber of the vector bundle and transport it along a trajectory of the flow. When the trajectory comes close to the starting poitn, we identify the fibers using the connection and we get a linear transformation of the fiber. The multiplicative ergodic theorem says that when the flow is ergodic a “matrix of mean monodromy” along the flow.

Moduli spaces of Abelian differentials.

Abstract version and a concrete version…..slides are pretty dense and mving a bit fast.

Teichmuller discs.

Teichmuller geodesic flow:

Teichmuller geodesic flow acts in the modulie space of pairs (complex structure, holomorphic quadratic differential.) Away from the zeros of a quadratic differential $q$ one can find a local coordinate $z$ on the underlying Riemann surface in which $q = (dz)^2$. This distinguished local coordinates defines a flat metric $|dz|^2$, which has a canonical singularities at the points where the quadratic differential has zeroes. Teichmuller geodesic flow acts as a uniform contraction in teh vertical direction and unform expansion in the horizongal direction.

(This is like Asteroids on a much richer surface than the 2-torus!)

The unraveled quotient space can be viewed as a polygon with parallel sides identified. There are some rich combinatorics available by cutting and regluing.

Hodge bundle and Gauss-Manin connection:

This reduces things down to $g-1$ Lyapunov exponents. To compute these exponents appear to be out of reach in most every dynamical system.

Siegel-Veech constant:

Closed regular geodesics on flat surfaces appear in families of parallel closed geodesics sharing the same lenght. Every such family fills a mximal cylinder having conical points on each of the boundary components. Denote by $N_{area} (S, L)$ the sum of areas of all cylinders spanned by geodesics of length at most $L$.

Theorem (Veech-Vorobets): For every $SL(2, R)$-invariant finite ergodic measure the following ratio is constant (ie.e does not depend on the value of a positive parameter L): $$ \frac{1}{\pi L^2} \int N_{area} (S, L) d\nu_1 = c_{area} (d\nu_1 ). $$ (The integration here is over the entire family of flat surfaces. Here $\nu_1$ is the invariant measure on the space of these surfaces.)

The constant $c_{area}$ is called the Siegel-Veech constant.

Eskin-Masur have a similar theorem for fixed $S$ where you take the limit $L \rightarrow \infty$.

What happens for the torus? For most of the tori, you can’t find closed small geodesics. However, inside the family of flat tori, there are some with very narrow cylinders with closed geodesics.

Eskin-Masur-AZ

Eskin-Okounkov computed the volumes explicitly.

Main Theorem: The sum of the Lyapunov exponents can be expressed as a sum of two (explicitly computable) constants.

V. Knizhnik

This stuff is amazing, truly beautiful. But, I can’t keep up with the typing….

Big advance by Eskin-Mirzakhani is in redaction.

Wednesday 05 October

Jacques Féjoz, Université Paris- Dauphine & Observatoire de Paris,Diffusion along mean motion resonance in the restricted three-body problem

(blackboard talk)

(joint work w M. Guardia, V. Kaloshin and P.Roldan)

This work would not exist w/o the marvelous 1964 paper of Arnold. This talk is closely related to the talk that Rafael gave yesterday and also to Vadim’s talk. I will try to concentrate on other aspects.

In the solar system, there is one priviledged place where we should look for instabilities. It is called the Asteroid Belt. It is located between Mars and Jupiter. Dust particles in this part of the solar system never condensed to form additional planets. Instead, there remain nearly 2 million particles ranging from microscopic to larger asteroids, some having a size of a few hundred kilometers in diameter. If you look to the current distribution of the asteroids in this belt, it gives a quite precise idea about the stability and instability zones between Jupiter and Mars.

In 1857, an American mathematician and astronomer named Daniel Kirkwood observed that there are gaps inside the belt where there are basically no asteroids and other zones where there are lots of asteroids. These gaps correspond to orbital resonances with Jupiter. Since the orbital frequency can be read off from Kepler’s third law….he draws a graph with vertical axis as the number of asteroids and the horizontal axis is the semi-major axis between the Mars and Jupiter radii. He highlights a gap appearing at the zone located in 3:1 resonance with the Jupiter orbit. Why are there these gaps? The conjectural explanation is that an asteroid is in resonance with Jupiter (which has a mass of about 1/1000 of the mass of the sun) then the eccentricity will be unstable. If the eccentricity goes through large variations, then its perihelon will be at size $a(1 - e)$ from the sun. Therefore, the asteroid will get closer and closer to Mars. Due to this very close encounter with Mars, the dynamics will transform so that the principal force acting on it will be due to gravity from Mars rather than with the sun. In this talk, I would like to focus on the first step in this scenario. Namely, why should the eccentricity vary a lot when the asteroid is in resonance with the Jupiter orbit?

Planar restricted 3-body problem: Sun, Jupiter, Asteroid. Restricted means we take the limit when $m_{asteroid} = 0$. Practically speaking, this means that we imagine the Sun and Jupiter take place along the 2 body motion and they are not influenced by the motion of the asteroid. Let’s normalize two things. Set $\mu = mass_{jupiter}$ and the mass of the Sun is $1 - \mu$.

There are 3 “small” parameters.

The mass of Jupiter $\mu \rightarrow 0$. The asteroid motion collapses then to an integrable 2 body problem.
The eccentricity $e_0$ of Jupiter. This parameter is slightly more subtle. The limiting dynamics as $e_0 \rightarrow 0$ is not integrable. We then obtain the circular restricted problem in which the two primary objects orbit on a circle centered at the center of mass. The problem restricts from 2.5 dof down to 2 dof.
Semimajor axis $a$ of the Asteroid. When we let $a$ go to zero, or to infinity, we encounter 2 body problems. In one limit, the Sun dominates the asteroid motion and Jupiter is irrelevant. In the other limit, the asteroid essentially sees the gravity of a combined mass of the Jupiter and Sun.

In the problem inside our solar system:

$ \mu = \frac{1}{1000}$
$a = (\frac{p}{q})^{2/3}$. This implies that the periods of the asteroid and Jupiter satisfy $ \frac{T}{T_j} = \frac{p}{q}$.
We will restrict attention to $0 < e_0 \ll 1$. This allows us to view the problem as a singular perturbation of the restricted circular problem. There is also a computational reason for doing this. Part of the proof will require some numerical computations. Our strategy was to make these calculations as simple and convincing as we possibly could. All the numerical computations are done on the circular problem and boil down to checking for zeros of a one variable function. A final reason is that the real eccentricity of Jupiter is $\frac{1}{20}$. There is hope that we could in fact claim our theorem for the real value. This will require some quantifications which in principle we could extract.

Theorem: Set $\mu = \frac{1}{1000}$. Fix $\frac{p}{q} = 7$ (chosen for incidental reasons; we expect this can be relaxed; so this asteroid is outside of Jupiter corresponding more closely with Uranus….nice discussion). Assume $0 < e_0 \ll 1$. There exists a solution and a time T with $e(0) < e_{min} = 0.48$ and $e(T)> e_{max} = 0.67$ and all the while the asteroid radius $a(t) \thicksim (\frac{p}{q})^{2/3}$. Thus we have a $(p:q)$ orbital resonance with Jupiter.

What is the time scale of $T$? Conjecturally, we have $T \thicksim - \frac{\ln \mu e_0}{\mu^{3/2} {e_0}}$.

What are the units of time? Year of Jupiter.

Circular Problem:

$$ H = \frac{|p|^2}{2} - \frac{1}{|q|} + \frac{1}{|q|} - [ \frac{1-\mu}{|q + \mu|} - \frac{\mu}{|q - (1 -\mu)|}]. $$ This formulation views the principal force as provided by a fictitious mass at the origin perturbed by the separation of the Jupiter and Sun masses.

Delaunay Coordinates (written in notation of Poincaré): $(L, l, G, g) $

$L = \sqrt{a}$
$G = \sqrt{a}\sqrt{1 - e^2}$ (angular momentum)
$g$ is an angle to Jupiter.
$l$ is the angle of the asteroid advanced past Jupiter.

If we set $\mu =0$, the perturbing term vanishes and we are left with a degenerate 2BP. Understanding this limit does not bring much light to the problem.

Assume $\mu >0$. The first natural idea is to average out the fast orbital angle remaining inside this Hamiltonian. This involves first making a change of variable. The averaging process leads to an integral. This is a transcendent process. However, when $e_0, e \ll 1$, you can use the Laplace coefficients to make some computations by hand. However, this computation is not much help because we are interested in proving diffusion in the eccentricities. This calculation does give some intuition which suggests that the perturbation looks generic and the degeneracy observed in teh $\mu =0$ limit is broken.

Fact (Numerical): There exists a normally hyperbolic cylinder foliated by periodic orbits $\gamma_e$, $0.48 < e < 0.67$. This is related to an idea of R. Moeckel from the late 90s. He draws some surfaces intersecting and explains that there are some small splitting issues requiring high precision arithmetics. We limited our attention to higher eccentricities to avoid these issues. Morally, the larger value of eccentricity the faster the diffusion. As we go above $0.67$ we get closer to the Euler relative equilibrium $L_{1,2}$.

In order to lower the number of dimensions, it is perhaps a good idea to consider a Poincaré return map to $[g = 0]$. This means we are looking at what happens every time the ellipse of the asteroid is aligned along the line connecting the Sun and Jupiter. He draws a helix of 7 levels high above an ellipse. Then he slices the helix with a vertical plane and identifies the intersection points as 7 normally hyperbolic invariant cylinders. It is now time to introduce the analogs of the “inner” and “outer” maps that Rafael introduced in the more general case yesterday.

OK, mostly pictures now….

Discussion: There is no steepness in this Hamiltonian so Nekoroshev’s theorem does not apply.

Boris Khesin, University of Toronto,Optimal transport and geodesics on diffeomorphism groups

(blackboard talk)

(joint work with J. Lennels, G. Misiolek, S. Preston)

Plan:

Euler equation on $SDiff$. Otto’s calculus.
$SDiff \subset Diff$ (numerous applications in optimal transportation)
$L^2$ and $H^1$ metrics.

I. Arnold’s approach to the Euler equation

This was discussed in Shnirelman and Brenier’s talks so I can perhaps be brief.

Definition: Consider $v$ to be a velocity field on a manifold $(M, (,))$ which is divergence free. The Euler equation is then given as $$ \partial_t v + (v \cdot \nabla) v = - \nabla p, ~ \nabla \cdot v = 0. $$ Here $\nabla$ is the covariant derivative. If there is a boundary, we assume that $v$ is parallel to $\partial M$.

Draws a picture and identifies the Lie algebra $g$ at the identity. He explains how to use Lie algebra transportation along a geodesic in $G$.

Theorem (Arnold, Bombshell in the 60s): The Euler equation may be viewed as a the geodesic equation on $G = SDiff (M)$ (the group of volume presernving differomorphisms) w.r.t right invariant energy $L^2$-metric given on $g = Lie(G)$ by $$ E(v) = \frac{1}{2} \int_M (v,v) \mu.

Remark: Other groups and energies give Euler top, Kirchoff equations for motion of rigid body in a fluid, KdV, Camassa-Holm, MHD, Landau-Lifschitz equation, …

II. Geometry of full diffeo group

This is the point of view rather common in optimal transport. This discussion unifies these two perspectives.

Consider $Diff$, the group of diffeomorphisms on $M$. Inside this group, we have the subgroup of volume preserving diffeomorphisms $SDiff$. The notion of $SDiff$ requires the volume form. Ebin-Marsden. He views $Diff$ as a space of fibers over the space of densities. Once you specify the volume form, this induces the fibration over the densities. Fibers $=F_\nu = [g \in Diff: g_* \mu = \nu]$.

$\exists$ “natural” $L^2$-type metric on $Diff$ for flat $M$: $$ l^2 [g(t, \cdot)] = \int_0^1 ( \int_M (\partial_t g, \partial_t g) \mu) dt. $$

Remark: $\forall ~M$,

$$ (v \circ g, v \circ g){L^2} = \intM (v \circ g, v \circ g)_{g(x)} \mu(x). $$

Properties:

Not right invariant on $Diff$
Is right invariant on $SDiff$, because the Jacobian term arising from the change of variables disappears.
“flat” for a flat $M$: $Diff \thicksim L^2 [ g(x)]$…pre-Hlibert.
geodesics in $Diff(M) \iff $ solutions of the Burgers equation: $$ \partial_t g (t,x) = v(t, g(t,x)); ~ \partial_t v + (v \cdot \nabla) v = 0. (*) $$ This differs from the Euler equation since the right side is zero. It also does not require the zero divergence condition.This formulation appears in the paper of Ebin-Marsden.
Geodesics which are orthogonal to $SDiff \iff $ potential solutions $v_0 = \nabla \phi$.

The proof of (*) follows from the chain rule and the fact we are considering geodesics. $$ 0 = \partial_t^2 g = \partial_t (v(t, g(t,x)))$$ $$ = (\partial_t v + (v \cdot \nabla v))(t, g(t,x))$$.

Remark: Geodesics on $SDiff$ are constrained within the larger family $Diff$ to remain on the subgroup. This requires imposing a force to keep the evolution within $SDiff$. This force is the pressure.

What I am describing right now is called Otto’s Calculus which arose in the study of optimal transportation.

Remark: It turns out there is a natural metric on the space of densities. We can introduce a measurement of the cost to move one density $\mu$ to another $\nu$. The natural metric is called the Wasserstein-Kantorovich $L^2$-metric on densities: $$ dist(\mu, \nu) = \inf_{g_* \mu = \nu } \int_M |x - g(x)|^2 \mu(x). $$ This is the cost of transporting $\mu$ to $\nu$.

Theorem (F. Otto): $$(Diff, L^2) \longmapsto (Densities, dist)$$ is a Riemannian submersion.

Corollary: Geodesics in the space of densities starting at $\mu$ are in 1:1 correspondence with horizontal geodesics in $Diff$ starting at the identity.

This is the picture behind the scenes driving the proofs of many theorems.

Applications:

Conjugate points along the base of densities correspond to focal points inside the space $Diff$. P. Lee, A. Agrachev and a former student (I missed the name…)
Asymptotic Directions $\iff$ geodesics with higher than ususal $(\epsilon^3)$-tangency. Theorem: Asymptotic directions to $SDiff$ must satisfy $$\nabla \cdot v = 0,$$ $$ \nabla \cdot (v \cdot \nabla) v = 0.$$ (These are called the Bao-Ratiu equations and arise naturally from this perspective.)

Theorem (K-Misiolek): For $M$ of dimension 2 (surface), ${\overline{K}} \neq 0, ~ \forall x \in M$ there do not exist asymptotic directions. (For $K > 0$ this result was called Palmer’s theorem.)

So asymptotic directions are “rare.”

What happens if we consider slightly more general metrics instead of $L^2$. Recently, there was interest in $H^1$ metrics so let me say a few words about that.

dimension	1	2	3
$SDiff$	Rot	$H(x,y)$	$[Jac = 1]$
Density	$[\hat{f}(x)]$	$[f(x,y)]$	$[f(x,y,z)]$

III. $H^1$-(right-invariant) metrics on $Diff$

article

$$ (v,v){H^1} = a | v |{L^2}^2 + b | \delta v^\flat |{L^2}^2 + c | d v^\flat |{L^2}^2 $$

where $ v \in Vect \rightarrow v^\flat \in \Omega^1 (M)$. So, we have terms involving $\nabla \cdot v$ and another involving $curl v$, etc.

Theorem: The Euler-Arnold equations are

$(n = 1)$

$b = 0 \implies$ Burgers (KdV for Virasaro)
$a = b = 1 \implies $ Camassa-Holm equation: $v_t - v_{txx} = -3 v v_x + 2 v_x v_{xx} + v v_{xxx} $\nabla \cdot
$a=0 \implies$ Hunter-Saxton equation: $u_{txx} = - 2 u_x u_{xx} - u u_{xxx}$

For other $n$, one can write the corresponding equation. Various other equations arise such as Euler-$\alpha$ and many others….

There exists one metric which has nicer properties than others.

IV $H^1$-metric on Densities

$(a = c = 0, b = \frac{1}{4} \neq 0)$. So, we are considering the metric $\| v \|_{\dot{H}^1}^2 = \frac{1}{4} \int |\nabla \cdot u |^2 \mu.$

Theorem: For any compact $M$ there exists an isometry $\Phi: Densities \rightarrow U \subset S_\rho^\infty$ (an infinite dimensional sphere) where $\rho = \sqrt{vol(M)}$.

Remark: The dimension 1 case was observed by Lennels. At first, we thought this was a special case but turns out to be general and produces some nice insights.

Geodesics on densities are great circles on the sphere.
They are solutions of a high dimensional Hunter-Saxton equation (completely integrable system)

Proof: $\Phi: \eta \in Diff \rightarrow f = \sqrt{Jac (\eta)}$. Then $$ \int_M f^2 \mu = \int_M (Jac (\eta)) \mu = \int_{\eta(M)} \mu = vol(M)$. $$ This very metric on the sphere arised earlier in probability theory and is known as the Hellinger distance, aka Fisher-Rao metric. All these objects come together from this point of view.

article

Bassam Fayad, IMJ CNRS,Smooth linearization of commuting circle diffeomorphisms

(blackboard talk)

My work was completely influenced by Arnold’s papers and ideas. I had very close friends who were his students. They were completely venerating hi. I like to think that I am representing them here. They are two. These students report that what they miss most is long discussions with Arnold who had patience and knowledge and he fills you with interest.

I work on small denominators.

Mixing tori. Impossible on $T^2$. He drew a cube representing a 3-torus. The statement was not so explicit.

He connects this to the original motivation of Kolmogorov which revolved around an interest in finding mixing.

Circle Diffeomorphisms:

$$ f(\theta) = \theta + \tilde{\alpha} + \phi(\theta) $$

$\rho (f) = \alpha$ is the (irrational) rotation number.

Denjoy theory. If $f \in C^2$ then $ f = h R_\alpha h^{-1}$ where $h$ is a homeomorphism of the circle.

What is the regularity of the homeomorphism?

Siegel. Arnold.

Arnold showed that if $\alpha$ is Diophantine and $f$ is close to $R_\alpha$ then $ f $ is analytic and $h$ is analytic.

What is diophantine?

$\alpha \in DC (\gamma, \tau)$ if $ |\alpha - \frac{p}{q}| > \frac{\gamma}{q^{2 +\tau}}$
Best approximations. $\| k \alpha \| = \inf_l | k\alpha - l|$. The sequence of best approximations is defined by $$ \| q_n \alpha \| < \| q \alpha \|$$ for all $ q < q_{n+1}, ~q \neq q_n$.

Linearized equation:

$\phi (x) = \psi (x + \alpha ) - \psi (x)$.

The global problem remained open and was conjected by Arnold. Even without the closeness condition, Arnold conjectured that the rotation number being diophantine was all that was required to ensure the analyticity of the homeomorphism.

Herman 1976 ($H$-class of numbers), Yoccoz 1981 (all Diophantine numbers)

If $f = h R_\alpha h^{-1}$ with $h$ a homeomorphism and $\alpha$ is irrational and you have $f \circ g = g \circ f$ then $g = h R_{\rho{g}} h^{-1}$. (My quotation of the Qualifiers might be wrong here….be CAREFUL….)

Why does commutation imply higher regularity, more rigidity? The idea emerges from a paper by Moser 1981 who proved KAM smooth linearization of $f,g$ commuting if $(\alpha, \beta) \in SDC$

$(\alpha, \beta)$ are SDC (Simultaneous Diophantine Condition) if $\max(|k \alpha|, | k \beta |) \geq \frac{\gamma}{k^\nu}$.

Applying these techniques, you can show: If $(\alpha, \beta) \notin SDC$ then $\exists ~ (f,g)$ commuting then $h$ is not absolutely continuous.

Theorem (K. Khanin, F): $(\alpha, \beta) \in SDC$ and $ f \circ g = g \circ f$ in $Diff^\infty \implies h \in Diff^\infty$. KF Paper: Annals 2009

Very clever pivots in a case-by-case analysis. Some pigeon holes. Make friends with your enemy.

Thursday 06 October

Chong-Qing Cheng, Nanjing University,One way to Cross Complete Resonance

Nice introductory discussion of Arnold Diffusion, placing the principal settings studied so far in context. Mentions that there is a “definition” of Arnold Diffusion in v3 of Arnold’s book.

Some nice pictures suggesting the mechanism.

A big issue to overcome is that there are uncountably many barrier functions. One way is to study the regularity. This will impliy the finiteness of the Hausdorff dimension.

Resonance path. KAM iteration at complete resonant point. Very nice pictures of the Aubry set!

Interesting discussion following the talk between Cheng and Mather, comparing their respective strategies.

James Ellis Colliander, University of Toronto,Hamiltonian PDEs

I spoke so I didn’t type.

Artur Avila, IMJ, CNRS, Global theory of one frequency Schrödinger operators

(blackboard talk)

This topic can be introduced in several ways. I try to present this work in a way that is connected to the work of Arnold.

KAM-persistence of quasiperiodic motion.

One theorem of Arnold: $f$ analytic diffeo of $T = R/Z$ orientation preserving has a rotation number $\rho$.

Theorem: If $\rho$ is Diophantine and $f$ is close to translation then $f$ is linearizable (analytically conjugated to tranlation).

He also makes a conjecture. This should be global. This means that the hypothesis “f close to translation” should not be necessary. This was proved by Herman in a breakthrough work introducing new techniques, and eventually completely resolved by Yoccoz. These results are now understood in a new framework called renormalization.

What is the situation in higher dimensions? Not every $T^2$ has a notion of translation number. Suppose we have a diffeo that has a rotation number and is close to translation. What can be said? In higher dimensions, the local theorem survives. Herman asked: which aspect of the global theorem will survive? It’s a paper of Herman with a very long title…

“Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnolʹd et de Moser sur le tore de dimension 2.”

Example:

$$(x,y) \longmapsto (x + \alpha, A(x) \cdot y)$$

$A(x)$ will be some projective action. In particular, I will imagine that $A(x) \in SL(2;R)$. So, I am viewing the second coordinate as an element of $PR^2$.

$A(x)$ is a matrix ($E - \lambda v(x), -1$) (top row) and (1,0) in bottom row. Here the parameters are $E \in R$, $v$ is a trig polynomial, $\lambda >0$.

Rotation number?

In the first slot, it is clear that the number is $\alpha$. In the second slot, it depends on all the parameters. It turns out that in this case, it is well-defined and is montonic wrt $E$. Diffeos of the circle have those regions called Arnold Tongues and there are similar structures here…rationality condition…draws a Cantor-like set. He draws a “vertical curve” in the $(E, \lambda)$ plane and along that curve we have $\rho = constant$.

Theorem (Herman):

$\lambda \ll 1$, KAM works
$\lambda \gg 1$, Lyapunov exponent $LE >0$. Independent of E

$(\alpha, A)^n = (n \alpha, A_n)$. $A_n (x) = A(x + (n-1) \alpha) … A(x)$.

$$L = \lim \frac{1}{n} \int \ln | A_n (x ) | dx \geq 0. $$

If $(\alpha, A)$ is conjugate to translation then $LE = 0$.

OK, so what is the obstruction to globalization? IS $LE$ the only obstruction to conjugacy?

$LE = 0$, continuity argument. Goldstein-Schlag, Bourgain-Jitormskaya

At the endpoint of the supremum of the good parameter, we can not have analytic conjugacy. What broke down? You might think it is just that we lose analyticity. But this turns out to not be the case because in this context topological conjugacy $\implies$ analytic conjugacy. So, it is possible to have $LE =0$ while losing even the topological conjugacy.

$A$ is analytic, extends to a neighborhood $[|\Im x | < \epsilon]$. $A_n (x)$ is defined on the same neighborhood. Maybe the LE changes a bit as we move off the real axis? You see that if $(\alpha, A)$ is analytic conjugate to translation then this kind of complexified LE $$ \lim \frac{1}{n} \int_{Im x = a} \ln | A_n (x) | dx = 0. $$ is still zero. Now, you have some kind of necessary condition that is implied by analytic conjugacy.

Theorem: $(\alpha, \rho)$ diophantine. $(\alpha, A)$ is analytically conjugate to translation $\iff ~ LE =0$ for $|\Im x|$ sufficiently small.

The method of proof of this theorem is quite interesting, but I won’t do that right now. You can’t just apply KAM theorem. It is closer to a different theorem. You need an a priori bound on the renormalization. In fact, you have that the complexified LE is behaving subexponentially…lots of work to be done to get conjugacy. What I want to do instead is to connect with the title of the talk.

$l^2 (Z)$

$ (Hu)n = u{n+1} + u_{n-1} + v(n \alpha ) u_n. $

$$ v: T \rightarrow R, ~ analytic. $$

$$ Hu = Eu. $$

This generates a one-parameter family of cocycles. $H \iff $ one-parameter family of cocycles. Those cocycles have been studied a lot for similar reasons why dynamicists should interested in cocycles. They are the simplest class with $LE>0$, are consistent with KAM theory, and have rich dynamical behavior (e.g transition from absolutely continuous and discrete spectra, etc.)

(spectrum should usually be thought to be a Cantor set, similar to the situation of those Arnold tongues.)

Local Theories:

When $v$ is small. This is often quantified by writing $\lambda v$ with $\lambda$ small. This was developed by Dinaburg-Sinai, Eliasson, Bourgain, Jitormiriskaya, Avila, Fayad, Krikorian, …. From the beginning, it started with $\lambda$ small so that you can apply KAM theory. An end conclusion here is that the spectral measures are absolutely continuous. sigh… What does this imply? You start with some state that lives in this lattice and you let it evolve. What happens here is that it spreads at the fastest possible transport. This is called ballistic motion, so it moves like the free problem.

Eliasson: not the same speed?….Avila:….in average over time, you can see that it is the same speed whenever there is continuous spectrum. Eliasson:….oh you time average, I see…. Craig:….seems you need some smoothnes. Avila:..(eagerly)….which kind of smoothness do you need? Craig:….need to integrate by parts. Avila:….I have some weak smoothness. Craig:….maybe can help. Avila:…I expect it will help but don’t know how to use it yet.

When $\lambda \gg 1$, there is a different theory. Sinai, Frohlic-Spencer, Eliasson, Bourgain, Goldstein, Schlag. This theory corresponds to the situation where $LE > 0$. (Typically…some almost every conditions), the spectral measure is pure point. This means that the infinite matrix is diagonalizable and the quantum dynamics is quasiperiodic. It is known that the eigenfunctions decay exponentially, so they are better localized than required by $L^2$. Two approaches to this: KAM and more recent interactive techniques like renormalization. With Bourgain, Goldstein, Schlag, new nonperturbative techniques developed based upon the assumption that $LE >0$. It might be possible to understand the dynamics across the entire parameter space.

He draws an egg. He draws a line along the egg representing strength of nonlinearity. When the nonlinearity is small, we KAM-like behavior. Both regions can be shown to be open. When the nonlinearity is big, we $LE >0$. The egg has an “region” in between. Does that intermediate region have a non-empty interior?

Natural global questions:

What is the behavior of typical one frequency Schrodinger operator?
In particular, is there an influence of other behaviors of cocycles?
You might be optimistic and hope to prove there is some kind of phase transition between the KAM-like and $LE >0$ regimes?
Describe the phase transition as “interface-like”? This would go in the direction of showing that there are not these other types of dynamics of cocycles.

This was basically blocked for some time. But, recently, well maybe not recently, it was in 2008, there emerged some new ideas to approach these questions. Large parts of the emerging program have been carried out.

Center your attention on $LE$ and its dependence upon parameters. It’s good to get some kind of target to focus your attention upon. There will be three regimes:

Supercritical: $LE >0$ (leads to localization; point spectrum)
Critical: otherwise
Subcritical: $LE = 0$ in a complex neighborhood $[|Im x| < \epsilon]$. (show that it is KAM-like; AC spectrum)

The main parts of the program. Study the critical “interface”. Study how it intersects one-parameter families and so on. This would be a kind of geometric approach to begin to understand the dynamics across the parameter space.

What parts are completed? Several parts…..here is a main theorem.

Theorem: $H$ is a typical one-frequency Schrödinger operator. $H = H_+ \oplus H_-$. $\sigma(H_+) \cap \sigma(H_+) = \emptyset$. $H_+ $ is supercritical, localization. $H_-$ is subcritical and KAM-like, AC spectrum.

What about the growth of critical energies? There are no critical energies. This is a bit surprising. As I said before, I had a picture moving up from the KAM region and we encounter a transition point. It would be natural to expect that you will face a critical energy. This turns out to not be the case. The critical interface has zero measure, a kind of Cantor set, inside a codimension 1 subspace. This means it won’t intersect a typical one-parameter family. All this takes place inside the geometric analysis of the parameter space.

$\alpha$ diophantine, subcritical is $KAM$-like. Even thought I don’t have the conjugacy, I still have good control on the dynamics. This is the concept of almost-reducibiilty.

What is the meaning of typical? $\alpha$ is almost every. We also have $v$. What’s a zero measure set in infinite dimensions? It involves some kind of concept called prevalence and Gaussian measures. My good set of $v$’s has the following property: $\forall ~ v_0, $ for almost every $\lambda_n \in [-1, 1]^Z$ then $v_0 + \sum \lambda_n^{\epsilon_n} e^{2 \pi i n \cdot x}$ where $\epsilon_n = \frac{1}{(n!)^2}$.

Mikhail Sevryuk, Russia Academy of Sciences, The reversible context 2 in KAM theory for lower dimensional tori

A review of KAM theory. “Meta-reason” for the ubiquity of invariant tori carrying conditionally periodic motions: any finite dimensional connected compact Abelian Lie group is a torus.

Various structures (contexts):

Hamiltonian
reversible
volume preserving
dissipative (no structure at all)

Key advances highlighted here.

Moser 1966
Herman 1988

Lai-Sang Young, New York University,Toward a smooth ergodic theory for infinite dimensional systems

(Scanned hand-written marker slides! Cool….we party like its 1999!)

Aim of larger project:

Extend finite-d nonuniform hyperbolic theory (=ergodic theory of chaotic systems) to $\infty$-d
New phenomena
include settings related to some PDEs. (principles should include nonempty set of PDEs.)

The body of finite-d stuff that I have in mind does not include Hamiltonian systems. Instead, we are looking at problems which include some dissipation. So the invariant sets we are looking at are like attractors, etc.

Today’s talk:

Reduction to finite-d via $W^c$-inf and $W^\epsilon$-foliations. Upshot: notion of “a.e.” initial conditions in $\infty $ dimensions.
Example of strange attractors from Hopf bifiurcation + forcing. Illustration of how to leverage finite-d techniques
Lyapunov exponents, periodic orbits and horseshoes for semiflows on Hilbert spaces (extend Katok’s results for finite-d diffeos.)

Some background info:

Givne an evolutionary PDE, view this as an “ode” on a function space. I want to see it as a dynamical system.

$$ \frac{du}{dt} + Au = F(u) $$ where $u \in X, ~ A $ is a linear operator, $F$ is the nonlinear part.

To define (smooth) dynamical system, need $(X, | \cdot |)$ such that

$u(0) \in X \implies u(t) \in X ~ \forall t \geq 0$,
$ t \longmapsto u(t), ~ t \geq 0$, continuous,
Smoothness of the time-t map $f^t : (X, \| \cdot \|) \longmapsto (X, \| \cdot \|)$ which maps $ u(0) \longmapsto u(t)$.

for nonunif hyperbolic theory, generally require $C^{1 + \alpha}$.

Model setting:

$X$ is a Banach space. $A$ is an operator on $X$. Assume $A$ is “sectorial” or self-adjoint w eigenvalues on $[a, \infty)$.

Q: Can I just think of $A = \Delta$? A: Yes. (OK, I’ll think that way….knowing that there are generalizations.)

A sample result:

Theorem (Henry ~80): …

Discussion….skip it….just know that we are not talking about an empty set.

“Solution” means mild solution.

I. Reduction via center manifolds

$W^c$ can be local, global, or “medium size”

Constantin-Foias-Nicolanenko 86, Chow, Sell, Mallet-Paret, Lu, …

Think of $f$ as the time one flow-map associated to this dynamical system on $X$.

(A1) Reference Splitting: $X = E^c \oplus E^s$, closed subspace, not invariant. (A2) Absorbing “slab”: $\forall ~ R ~ \exists ~ R’$ such that $f( E^c \times B^s (0,R)) \subset E^c \times B^s (o, R’)$. (A3) INvariant cones ….lots written on slide here, can’t keep up with that. Some nice pictures to explian what is going on.

$E^s$ is vertical, $E^c$ is horizontal.

….questions….is the center manifold infinite dimensional?…..this is just the setting. I’ll be precise about the theorems soon.

some spectral assumptions.

Theorem 1 (Existence of $W^c$): $\exists ~ ! ~ W^c = graph(h^c), ~ h^c : E^c \rightarrow E^s, ~C^{1+\alpha}$, invariant.

Theorem 2 (Existence of $W^s$ foliations): slide slid up…..

Theorem 3 (Absolute continuity of $W^s$-foliations): (Zeng Lian, Chongchun Zeng, LSY): Assume $dim(W^c) < \infty. Then $W^s$-foliation is absolutely continuous.

Strange Attractors arsising from periodically forced Hopf bifurcations.

(joint work w. Kening Lu and Qiudong Wang)

Result for ODE in 2D $\rightarrow $ Corresponding equation for evolution equation in Hilbert space $\rightarrow $ Application to a specific PDE.

We have an unforced system with a parameter which is undergoing a “generic” supercritical Hopf bifurcation at $\mu = 0$.

$\mu < 0$
$ \mu = 0$
$ \mu > 0$.

Normal form. Introduce the twist number, expressed in terms of coefficients appearing in the normal form.

Forced system. Periodically, we kick it and then let it relax. It doesn’t have to be a kick. It just needs to relax in between the applications of the forcing.

Theorem (LWY): …I read it rather than type it…. there is some number you can calculate that as to be pretty big. We have a big kick period. Then you have a strange attractor with complicated dynamics. The attractor has an SRB measure.

An SRB measure is an important concept in finite-d and is the first challenge to bring it to infinite dimensions. If you look at a Hamiltonian system with flowmap $\phi_t$. Let $m$ be the Liouville measure. Assume ergodic. Then $\forall$ cts $g$ we find $$ \frac{1}{T} \int_0^T g( \phi_t) x dt \rightarrow \int g dm $$ for $m-a.e.$ x. (Birkhoff Ergodic Theorem)

Now suppose you have an attractor. (Sinai-Ruelle-Bowen). An invariant measure $m$ is called SRB or physical measure if the same convergence takes place for Lebesgue-a.e. In this setting $m$ is completely singular compared to the ambient Lebesgue measure. This is considered to be the analog of the Liouville theorem for dissipative systems.

I am claiming that these attractors support these measures.

Example, nice pictures.

Kick can be quite general.

…

Lyapunov exponents and $W^u, ~ W^s$-manifolds

Ruelle, Mané, Thieullen 80s, Lian-Lu (later)

Cocycle set up. Biggest differences w. finite-d:

$\Phi (x)$ generally not onto (possibly 1:1)
“Essential spectrum” - Lyapunov exponent is not defined.

Kuratowski measure of noncompactness.

Extension of Katok’s results….moving a bit fast here.

Friday 06 October

(Alas, my flight departure time will force me to miss out on hearing these talks.)

Laurent Stolovitch, CNRS, Université de Nice, Smooth Gevrey normal forms of vector fields near a fixed point

Claude Viterbo, Ecole Normale Supérieure, Symplectic Homogenization

Anatoly Neishtadt, Loughborough UniversityAveraging, passages through resonances, and captures into resonance in dynamics of charged particles

(Image no longer available: Anatoly Neishtadt)

Île de Berder Workshop Notes

Fri, 09 Sep 2011 05:22:49 +0000

Île de Berder Workshop Notes

I am at a workshop on Île de Berder. The post below contains the notes I am taking during the talks. I apologize (especially to the speakers) for misquotations and typos but I hope the notes might be useful.

Tuesday 2011-09-06

Rafik Imekraz: Non resonant normal form for perturbed quantum harmonic oscillator

Study high sobolev norms of solutions $\psi$ solving a NLS with harmonic trap and a nice smooth bounded potential. There is also a compact operator $M$ which is introduced to avoid the resonance. The natural Sobolev spaces are those naturally associated to the linear operator (without the compact operator).

Almost global existence. $\exists ~M$ with small operator norm and for data of size $\epsilon$ in $H^s$ then the solution stays small in that norm for times on the order of $\epsilon^{-3}$.

Earlier results on NLS and NLW on $S^1$. Delort-Szeftel 2007, Bambusi-Delort-Grébert-Szeftel on Zoll Manifolds.

In his thesis he replaced the $x^2$ by $x^{2p}$, e.g. the quartic oscillator.

When $V=0$, we have the usual Hermite eigenfunctions.

This PDE can be given a Hamiltonian formulation. The operator $M$ is given as a Fourier multiplier on the eigenbasis with $m_j \rightarrow 0$.

Normal Form Procedure:

$H_0 + P $ transforms into $H_0 + Z + R$ with $Z$ in normal form and $R$ a negligible error term.

Spectral key points. $H_0$ has a nonresonance condition. We will say that the spectrum is nonresonant. Definition introduced initially by Bambusi, used by Delort, Brebert, Imkeraz, Paturel, Szeftel. The condition involves infinitely many eigenvalues.

Delort-Szeftel argument. The spectrum is not explicit when $V \neq 0$. However, there is a 2005 theorem by Klein-Korotyaev-Pokrovski: $$V \in BC^\infty(\mathbb{R}, \mathbb{R}) \cap L^1 (\mathbb{R}) \implies |\lambda_j - 2j -1| \leq \frac{C}{j^\delta}.$$ Same assumption on a Zoll manifold.

We have a multilinear estimate. Technical proof with a commutator lemma. Uses smoothness and $L^p$ estimates on eigenfunctions by Yajima-Zhang 2001. It is not necessary to know the eigenfunctions explicitly.

Conclusion: A normal form procedure is possible to deduce almost global existence for $$ i \partial_t \psi = ( - \partial_x^2 + x^2 + V(x) + M) \psi \pm |\psi|^2 \psi. $$

Is it possible to say something when $M=0$? Does there exist a $V$ which has a nonresonant spectrum? We give a partial answer. Yes, there exists such a $V$ but without the regularity we’d like to impose. The function $V$ will be continuous and bounded but we don’t know if it is possible to create a more regular $V$ which remains nonresonant. The lack of regularity seems to preclude the multilinear analysis.

Chelkak-Kargaev-Korotyaev 2004

Q: Does the proof of almost global existence imply global well-posedness with polynomial-in-time bounds on the high Sobolev norms?

There was some discussion but the answer was not clear. It turns out the question was naive because there are examples showing blowup.

Massimiliano Berti: Quasiperiodic solutions of Hamiltonian PDEs

Hamiltonian PDES.

Goal: existence of qp solutions of pdes. techniques, based on nash-moser implicit function theorem and KAM theory. Techniques apply to NLW, NLS and 1d derivative-NLW.

Perspectives: water waves?

Model case:

$$u_{tt} - \Delta u + V(x) u = \epsilon f(\omega t, x, u)$$

$x \in T^d$, periodic boundary conditions
$\epsilon$ small
$V(x) \in C^k (T^d; R)$
$f \in C^k$

NLW is a Lagrangian equation on an infinite dimensional phase space.

The problem is to construct qp solutions of the NLW for $\epsilon \neq 0$.

QP Definition: $u(\omega t, x)$ where $u(\phi, x): T^\nu \times T^d \rightarrow R$

Linear equation.

superposition principle….harmonic oscillator with frequence $\sqrt{\lambda_j}$ when $\lambda_j >0$. For $\lambda_j < 0$, we have a harmonic repulsor.

There are infinite dimensional spaces of qp solutions of the linear equation. We want to see if these persist for the nonlinear equation when $\epsilon >0$ is small.

The embedding $T^\nu \ni \phi \longmapsto u(\phi, x)$ solves the “NLW” $$ (\omega \cdot \partial_\phi)^2 u - \Delta u + V(x) u = \epsilon f(\phi, x, u). $$ in a Sobolev space $H^s (T^\nu \times T^d )$.

This can be approached as a bifurcation problem. Let $F(\epsilon, u)= (\omega \cdot \partial_\phi)^2 u - \Delta u + V(x) u - \epsilon f(\phi, x, u).$ We know there are zeros when $\epsilon =0$ and we want to branch off these via the implicit function theorem. The standard hypotheses of the implicit function theorem are not satisfied.

We need to make a diophantine assumption to proceed. We use Newton Method + “smoothing” following the Nash-Moser IFT.

Newton tangent method for zeros of $F(u) = 0 + “smoothing”$: $$ u_{n+1} = u_n - S_n (D_u F)^{-1} (u_n) F(u_n), $$ where $S_n$ is a regularizing operator.

Advantage: Quadratic scheme! $$| u_{n+1} - u_n |s \leq C(n) | u{n1} - u_{n-1} |_s^2. $$ This is convergent even when the constants $C(n)$ are exploding.

However, there are also disadvantages. We are studing a linearized equation on an approximate solution. Linear differential operator with non-constant coefficients. It is not diagonal in Fourier basis. We know the eigenfunctions exist and are orthonormal in $L^2$ but we don’t have much explicit control. This is a “singular” perturbation problem.

Literature:

Kuksin 89; Wayne 90: Dirichlet b.c., $f$ analytic, KAM theory. Eigenvalues of $- \partial_x^2$ are simple so the KAM nonresonance conditions are satified (so-called 2nd Melnikov conditions)

Craig-Wayne 93: Periodic case. Eigenvalues have multiplicity 2. Lyapunov-Schmidt, f analytic, Newton-Method, periodic solution. Extended to PDE nonresonant of Lyapunov center theorem. …breath mention by Bourgain.

Berti-Bolle DMJ 06, Advances 08. Berti (book) 08. Extend to PDE the Weinstein-Moser and Fadell-Rabinowitz theorems.

In the space dimension $d \geq 2$, main difficulties:

eigenvalues of $-\Delta + V(x)$ appear in clusters of increasing size. For example all the lattice points on spheres have the same linear frequency.
Feldman-Knonner-Trubowitz. The eignefunctions of $-Delta + V(x)$ are NOT localized with respect to exponentials. This means that there are strong interactions between the eigenmodes. In this frame, it is often convenient to work with pseudo-PDE involving Fourier multipliers. (Bourgain, Kuksin-Elliason)
Bourgain 95-98 f analytic.
Bourgain’s question 97: for differentialble nonlinearities? See Berti book.
Berti-Procesi DMJ 11, General Riemannian manifolds. General Lie group: products of eigenfuctions can be represented as a sum over eigenfunctions. Related to Birkhoff normal form results by Bambusi, Delort, Grébert, Szeftel.

QP solutions for $d \geq 2$:

Newton Method. Bourgain Annals 98 ($d=2$); Annals 05; Wang 11 Supercritical (completely resonant) NLS-NLW, no parameters.
KAM Method: Kuksin-Eliasson Annals 10. analytic NLS w Fourier multipliers.
Procesi-Xu 11, Procesi-Procesi 11, any dimension, Birkhoff normal form for completely resonant NLS.

New results for qp solutions in $d \geq 2$:

Berti-Biasco CMP 2011
Bambusi-Berti-Magistrelli JDE 2011.

Techniques:

Optimal Nash-Moser iterative scheme: different from the analytic Newton iteration.
For measure estimates, we use simpler techniques that Bourgain avoiding semi-algrebraic sets.

Very interesting technical discussion highlighting the favorable constellation effects when the frequencies are in tight resonance.

Granville: More “torsion” of a manifold there are less integers nearby. (This seems interesting…look up and discuss with Andrew.)

Wednesday 2011-09-07

Frédéric Bernicot: Bilinear Strichartz inequalities and space-time resonances

(joint work with P. Germain)

Linear Strichartz inequalities for Schrodinger. $L^2$ admissibility pairs. History: Strichartz 77, Ginibre-Velo 89, Keel-Tao 98. Extensions to compact manifolds with loss of derivatives.

What about bilinear Strichartz inequalities? Suppose $f \longmapsto e^{it \Delta} f = u, g \longmapsto v$. We’d like to know:

$$ | vw |{L^p L^q (R^{1+d})} \leq |f |2 | g |_2. $$

In previous works on this topic, these were typically studied with $p, q$ both equal to 2. We are interested in the cases where p and q are not equal to 2.

Applications: some large time behavior results; some stability results.

Time resonant set; space resonant set; their intersection is called the spacetime resonant set.

Take advantage of geometric properties of the resonant set to prove boundedness properties for solutions.

Some related works could be mentioned:

The detailed study discussed here also resonates with the paper of Bejenaru-Herr-Tataru 2009:A convolution estimate for two-dimensional hypersurfaces.

Emanuele Haus: Asymptotic stability of the synchronous resonance for an elastic satellite with internal friction

(joint work with D. Bambusi)

Synchronous resonance: the satellite always shows the same face to the planet. For example, the moon does this. Why does this happen? Tidal effect. The satellite is deformed and stretched towards the planet. If the satellite is not in a circular synchronous orbit, the direction of the stretching changes inside the satellite –> dissipation. Our aim is to stydy the system of coupled equations and prove asymptotic stability of the synchronous resonance. We want to model the orbital, rotational and internal degrees of freedom.

Internal friction –> circular orbit + 1:1 resonance is a (local) attractor.

Spherical case was done earlier by D. Bambusi.

J. Colliander: Normal Forms and the Upside-Down $I$-method

(joint work with Soonsik Kwon and Tadahiro Oh)

Rémi Carles: Interaction of coherent states for Hartree equations

This talk will involve semiclassical analysis and the equation might not be Hamiltonian.

Schrodinger equation in semiclassical regime ($\epsilon \ll 1$): $$ i \epsilon \partial_t \psi^\epsilon + \frac{\epsilon^2}{2} \Delta \psi^\epsilon = V(x) \psi^\epsilon. $$ We study semiclassical wave packets (coherent states), $$ \psi^\epsilon (0, x) = \frac{1}{\epsilon^{d/4}} a ( \frac{x-x_0}{\sqrt{\epsilon}}) e^{i p_0\cdot (x-q_0)/\epsilon}, ~ a \in S(R^d). $$ The potential $V$ is smooth, real-valued and at most quadratic: $ V \in C^\infty, \partial_x^\alpha V \in L^\infty, ~\forall |\alpha| \geq 2. $$ (No sign assumptions) The associated Hamiltonian flow is globally well-posed but might involve exponential growth.

Classical action: $ S(t) = \int_0^t (\frac{1}{2} |p(s)|^2 - V(q(s))) ds. $

Equation for $\psi^\epsilon$ is equivalent, via algebraic manipulations to $$ i \partial_t u^\epsilon + \frac{1}{2 } \Delta u^\epsilon = {\mathcal{V}}^\epsilon u^\epsilon. $$ where $ {\mathcal{V}}^\epsilon =$…ack slide switch.

Ehrenfest time. Validity of the approximation with $V$ is not a polynomial. The difference between the appoximation and the original solution solves an inhomogenous Schrodinger equation. This is studied via energy estimates.

He carries ont a derivation of the ansatz from “scratch” by comparing things at different levels of $\epsilon^j$. The analysis “explains” why this is a reasonable choice.

Hartree equation. Same equation as before with additional term $(K * |\psi^\epsilon|^2) \psi^\epsilon$. Here we need $K \in W^{\infty, \infty}$. The ansatz is adjusted by adding in a new prefactor $\epsilon^\alpha$. The importance of the nonlinearity emerges differently depending upon the value of $\alpha$.

Two initial coherent states. When $\alpha = 0$, the Hamiltonian flow is affected by nonlinear effects. Modified $\epsilon$-dependent actions.

Hmmm…..I should recast interction Morawetz in the semiclassical setting and see if an interesting estimate emerges in the semiclassical limt.

Main Result: The exact solution can be approximated by two coherent states but there is a required phase drift between the coherent states. There is a corollary about the Wigner measures. The Wigner measure does not see the nonlinear effect except when $\alpha =0$.

Tiphaine Jézéquel: Homoclinic orbits with many loops near a $O^2 i\omega$ resonant fixed point for hamiltonian systems

(joint work w. Patrick Bernard and Eric Lombardi)

$$ \frac{du}{dt} = L_\epsilon u + Q_\epsilon (u). $$

$u \in R^4$
$t \in R$
$u=0$ is a fixed point, i.e. $Q_\epsilon (0) = 0$.

This implies that the dynamics in a neighborhood of zero. The resonance configuration is captured by the $O^2 i\omega$ resonance. 4 eigenvalues on imaginary axis, 2 above real axis, 2 below.

Physical Context

Motivated by study of water waves. 3d gravity-capillary fluid modelled by the Euler equation and we look for 2d traveling wave solutions. The “spatial dynamics method” produces an infinite dimensional equation. Using the center manifold theorem, this problem is reduced to a 4-d invariant manifold. We look for particular soltutions in the manifold. This is the collapse to 4d.

Initial aim: existence of solitary waves. In the R4 setting, this corresponds to a homoclinic connection to 0 in the center manifold. This turned out to be hard. So, we transferred to a different study. We study the existence of generalized solitary waves. This corresponds to a homoclinic connecton to a periodic solution.

Lombardi 2000.

Beautiful pictures. Excellent exposition of the phase space portraits in R4. Pictures are getting even better.

OK, the strategy was nicely described. The last part of the talk begins to show how the nice pictorial overview of the proof strategy is actually implemented. The details look formidable involving KAM, lots of ODE manipulations. She quotes ideas from Moser 1958, Russman 1964.

(I learned later from Tiphaine that she created her figures using Adobe Illustrator.)

Thursday 2011-09-08

Sandrine Grellier: Integrable effective dynamics for a nonlinear wave equation

(joint work with Patrick Gérard; arXiv preprint)

Consider the half wave equation:

$$i \partial_t u - |D| u = |u|^2 u$$

Here $|D|$ is what you expect. This is a tyo model for NLS on degenerate geometries leading to a lack of dispersion. This equation admist the same kind of conservation laws as NLS:

$H(u)$
$p(u)$
$Q(u)$

Compared to the $NLS_3^+$, this is a nondispersive equation..

$\Pi_+$ is projection onto Fourier modes $k \geq 0$. $\Pi_-$ is the projection onto modes $k<0$.

The equation is $L^2$ critical but the first iteration of the Duhamel formula is not bounded in $H^s$ for $s< \frac{1}{2}.$ Despite this, we have a Cauchy theory in $H^{1/2}$. There exists a unique solution in $C(R; H^{1/2}(T))$. Persistence of regularity also holds for $s>1/2$. The proof uses some Brezis-Gallouet type logarithmic inequalities. It provides rather bad large time estimates:

$$ | u(t) |{H^s} \leq e^{e^{Cs t}}. $$

We compare this to the cubic Szegö equation: $$ i \partial_t u_+ + \partial_x u_+ = \Pi_+ (|u|^2 u). $$ Decoupling result: Assume that $\Pi_+ u_0 = u_0$….ack slide changed….

Theorem: Let $s>1$. Assume that $\Pi_+ u_0 = u_0 = O(\epsilon)$ in $H^s$. Then, $u =v + O(\epsilon^{3-\alpha})$ on a time scale of size $\epsilon^{-2} \log \epsilon^{-1}$ where $v$ solves the half-wave equation. Furthermore, the solution remains small in $L^\infty$.

Corollary (Weak Turbulence): Let $s>1$. There exists a sequence of data $u_0^n$ and a sequence of times $t_n$ such that the data converges to zer in $H^r$ for all $r$ while the $H^s$ size of the solutions at time $t_n$ exceeds a power of $\log \frac{1}{| u_0^n |_{H^s}}$.

Contrast this with the 1d cubic NLS. Zakharov-Shabat 1972: no such norm inflation. For 2d cubic, CKSTT 2010 construct small $H^s$ data which grows large.

The proof comes from the “weak turbulent property” of the cubic Szegö equation. If the approximation result in the Theorem held on a longer time interval, we could prove a stronger weak turbulence result for the half-wave problem, more analogous to the corresponding result for Szegö where divergence to infinity has been established.

Quick sketch of the sequential norm inflation property for cubic Szegö. This follows from a rather explicit analysis of solutions of the form constant + pure exponential.

Sketch of proof: Analysis similar to what I spoke about yesterday. Nonlinear term is explicitly represented in terms of Fourier coefficients.. The $L^4$ expression is separated into the positive and negative frequency components in $L^4$ plus another term related to the $L^2$ norm and another term.

Removal of trivial resonances with a change of phase, as in Bourgain. Birkhoff normal form transformation. Resonance set is identified and has some algebraic structure so that the resonant quartets can be identified. There is no problem with small divisors here. This leads to a new system after these transformations. Our task is to show that this transformed system is approximated well by the Szegö equation. Smallness in $H^s$ can be shown via bootstrap on a time interval of size $\epsilon^{-2} \log \epsilon^{-1}$.

Lax Pair and a priori bounds for Szegö: Hankel operator….Lax 1968, Gérard-Grellier 2010. Peller 1982 shows that the trace of the Hankel operator is equivalent to the $B^1_{1,1}$ norm. This space is an algebra which contains all the $H^s$ spaces and is a subspace of $L^\infty$. The solution stays bounded in $B^1_{1,1}$.

There are many things to understand. We would really like to understand NLS on the Heisenberg group.

Nice discussion at the end of the talk explaining how the half-wave problem is sort of in between the Szegö equation and the NLS on the Heisenberg group.

Dario Bambusi suggested that a normal form iteration method a la Bourgain might allow for an improved approximation result.

Oana Pocovnicu: The Szegö equation seen as the resonant dynamics of a nonlinear wave equation

Similar to the last talk. My study will be done on the real line, rather than the torus. Why do that? Normal form methods work nicely on the torus. In the setting of the real line, we can still have small divisors. Cutoffs like done in Jim’s talk would create other issues and the approximating result would involve an equation other than the Szegö equation. Instead of using the Normal forms approach, we use the renomalization group (RG) method.

SE: $ i\partial_t u = \Pi_+ (|u|^2 u).$

In the case of the real line, we have one solution with initial data of an explicit form, then we can prove that the Sobolev norms behave like $$ | u(t) |_{H^s} \thicksim t^{s-1}. $$

Theorem: Let $v$ solve the half wave equation with initial data $\epsilon W_0$ where $W_0 \in H^s_+ (R).$ Let $u$ denote the evolution from the same data under the Szegö equation. Assume that $| u(t) |_s \leq C \epsilon (\log \frac{1}{\epsilon^\delta})^\alpha$ for $0 \leq \alpha \leq \frac{1}{2}$ then we have a good approximation….ack slid changed.

Then we have a corollary which transports the weak turbulence property of Szegö over to the half wave equation.

Remark: In order to show arbitrarily large growth of the solution, we need a better approximation result for a time of size $\epsilon^{-2 - \beta}$ where $\beta >0$. The point here is that the Szego equation is the resonant subsystem sitting iside the half-wave equation, analogous to the way the Toy Model sits inside cubic NLS on $T^2$.

RG method: Chen-Goldenfeld-Oono 1994, De Ville, Harkin, Holzer, Josic, Kaper; Ziane Temam, Moise, Abou Salem.

How does this method work?

We make a change of variable to remove the $\epsilon$ prefactor in front of the data and to move into the interaction representation. This introduces an equivalent equation with some exponentials and $\epsilon^2$ in front of the nonlinearity. We make a naive perturbation expansion in powers of $\epsilon$ and a Taylor expansion of the nonlinearity in terms of the unknown $w$. We plug and chug to identify powers of $\epsilon$. There are no ad hoc assumptions….we can just try….so it has some flexibility over the Birkhoff normal form.

The manipulations allow us to identify resonance as a vanishing of the phase function inside the Duhamel integral. It gows in time as a secular term and will cause our approximation to break down. We consider then the renomalzation group equation. We define a new approximating object which includes the explicit secular term.

Many resonances in this half-wave equation. The resonant set of the half wave equation ${ \phi (\xi, \xi_1, \xi_2, \xi_3) = 0}$ has non-zero measure for fixed $\xi$. Nice remarks connecting the resonant set to the discussion from Bernicot’s talk.

She has also obtained a second order approximating equation to the half-wave equation. This equation is Szegö plus some 5-linear terms. The approximation degree is tighter ($\thicksim \epsilon^5$) but on the same time interval $\epsilon^{-2} \log {\frac{1}{\epsilon}}.$ This quintic extension of Szegö is not yet understood.

Zaher Hani: Long time strong instabililty and unbounded orbits

Consider cubic NLS on $T^2$. This problem is LWP in $H^s$ for $s>0$ and GWP for $s>2/3$. We are interested in dynamical aspects of global flow. How do the orbits behave in $H^s$? Are all the orbits bounded or doe they grow in time? Such questions are crucial in understanding the frequency dynamics of the energy.

Upper bounds on the $H^s$ norm. WE have ocnservation of mass and energy which give a prior bounds on $L^2$ and $H^1$. There are no unbounded orbits in $H^s$ for $s =0, 1$. The 1d analog possesses infinitely many conservation laws that control integer Sobolev norms. Polynomial-in-time bounds have been established by Bourgain, Staffilani, CKSTT< DSPST, Sohinger, CKO. We don’t expect poly bounds to be sharp.

Lower bounds Does there exist a global solution fo cubic NLS that satisfies $\sup_{t \in R} | u(t) |_{H^s} = + \infty$?

Conjecture (Unbounded orbits conjectore): For $s>0, s \neq 1$ there exist global solutions to cubic NLS on $T^2$ that satisfy $\sup_{t \in R} | u(t) |_{H^s} = + \infty$.

Think of the solution supported on three frequency scales: low, medium, high.

High frequences need to become larger. Medium frequencies have to decrease to balance the increase at high frequencies. Conservation of mass requires that the low frequencies become larger to compensate for the net loss of mass at medium and high frequencies.

Theorem (CKSTT 2008): $\exists$ solution of cubic NLS which is initially small in $H^S$ but at some later time the solution exceeds an arbitrarily large size.

Caution: This does not imply the existence of an unbounded orbit.

Another related result is due to Carles-Faou.

Long-time strong instability: $X$ banach space. $S(t)$ is a continuous dynamical system on $X$. CKSTT result shows long time strong instability near the zero solution. This notion generalizes the CKSTT conclusion around a point other than the zero solution.

Lemma (H 11): Suppose $D \subset X$ is dense. If $S(t)$…ack slide changed…

Lemma suggests that to prove existence of unbounded orbits, it is enough to prove that LTS instability holds near a dense subset of $H^s$.

While proving that generic orbits are unbounded seems ambitions, we can formulate a localized version of the lemma. We don’t strive to prove the genericity of unbounded orbits. The localized lemma recasts the lemma above onto a closed subset $F \subset X$.

The proof is a straightforward application of the Baire category theorem. This is the program. It remains open whether this approach applies to NLS. Instead, we will obtain results on some other systems inspired by NLS. We make a first nontrivial step towards the implementation of this program for the cubic nonlinearity.

Theorem (LTSI near single-frequency data H 11): NLS exhibits LTSI near $A e^{inx}$ in $H^s$, at least for $s \in (0,1)$.

Consier NLS with a trilinear Hamiltonian. In the limit $R \rightarrow \infty$ the nonlinearity $\mathcal{N}_R \rightarrow |u|^2 u $ and we have the LTSI property for this sytem.

NLS in Fourier space. Recasting NLS in Fourier space following CKSTT. Parallelogram of four frequencies is required to excite activity at a frequency $n$. Resonance corresponds to the requirement that the parallelogram be a rectange. The restriction of the 4-wave interactions to the rectangles is called RFNLS. If the initial data are supported on a subset $\Lambda \subset Z^2$ and for any three vertices $n_1, n_2, n_3 \in \Lambda$ on a rectangle, we will say that $\Lambda$ satisfies the closure property if we are certain that the fourth vertex is also in the rectangle.

A rectangle is a first example of a set which satisfies the closure property. Consider the rectangle (0,0), (N,0), (0,N), (N,N). We can calculate explicitly the associated ODE system. Suppose that at time zero, we have the mass cocnetrated mostly at (N,0) and (0,N) while there is a little bit at (0,0), (N,N). At a later time, the mass moves across to the other diagonal. Thus, the Sobolev norm increases by a factor $2^{s-1/2}$. The CKSTT construction is a concatenation of this idea.

Step 1: Build a set $\Lambda = \Lambda_1 \cup \dots \cup \Lambda_P$ satisfying structural and geometric properties and the norm explosion property.

Step 2. Construct a solution to RFNLS.

Step 3. Approximation result. RFNLS approximates FNLS.

Step 3 is the easiest one in the CKSTT paper. Recall that the passage from FNLS to RFNLS involved throwing away the nonresonant terms. An integration by parts argument allows CKSTT to show that these terms contribute very little to the FNLS dynamics. These observations are the key steps to prove the approximation Step 3.

When the ground solution is changed from zero to a pure single frequency data, we have to show that the complete solution $u(t)$ is approximated well by the ground solution evolution plus the (adapted) CKSTT solution $v(t)$. By galilean invariance, we can assume that $n=0$ and $u_0 = A$ so that $u_g (t,x) = A e^{i |A|^2 t}$ and we would like to limit the interactions between the zero frequency and those in the resonant set $\Lambda$. Nonresonant interactions with the zero mode create problems. He calls this effect a second order resonance. This analysis identifies why the theorem is restricted at this stage to $s \in (0,1)$.

We define the nonlinearity $\mathcal{N}R$ by throwing away all interactions for which $|\omega_4| > R$. $\mathcal{N}_0$ is the resonant nonlinearity we saw before.

$R$-closure, a natural generalization of the closure property…..wow I like this! There is more flexibility in the CKSTT construction than I realized.

Berti’s Question: How fast is the diffusion? Answer: You have to track it through CKSTT. You will find this is a four tower exponential. Therefore, the rate of growth suggested by the CKSTT example is about $\log \log \log \log t$ which is pretty slow….but does diverge.

Nice discussion afterwards speculating on applications of the Baire Category lemma to the periodic Szegö equation.

Erwan Faou: 2d Cubic NLS: Energy cascades vs. Sobolev stability of plane waves

(reporting on two joint works, one with Rémi Carles another with C. Lubich and Gauckler)

This work was inspired by some numerical simulations.

Cubic NLS on $T^2$. We won’t care if it is focusing or defocusing.

Rewrite the Hamiltonian wrt Fourier coefficients.

When we add a convolution potential (diagonal in Fourier), we can prove stability results for small intial data. Bourgain, Kuksin, Craig-Wayne, Poschel, Eliasson-Kuksin, Bambusi-Grébert, Faou-Grébert. Typical results imply preservation of the actions for a polynomial (or slightly longer) time in terms of the size of $\epsilon$, the data size.

Without potential in $d=1$, there are stability results. Complete integrability.

Without potential in $d=2$. Many small quasiperiodic solutions. CKSTT instability result (which was mentioned in all the talks today!)

Semi-discrete system:

Space approximation.. We use a Fourier pseudo-spectral collocation method. We look for a trigonometric polynomial satisfying NLS at the grid of $K^2$ points. There is an aliasing problem.

Splitting schemes. We use a symplectic integrator.

The numerical is very close to the dynamics of the modified energy. Two instability mechanisms: small divisor issue and the aliasing problem. To avoid the small denominators we use a Courant-Friedrich-Lwey condition $\tau K^2 < C$. The alsiasing is avoided by making sure that the frequencies remain localized. (See Dario’s talk tomorrow on the stabiilty of solitons.)

Numerical tests on NLS. Generic prservation of the actions over extremely long times for small initial data. Typical picture of $\log |\xi_j (t)|^2. The graph consists of horizontal lines. When we start with 5-mode data, we see some more interesting dynamics.

Energy Cascade (with Rémi Carles)

arXiv: Carles-Faou

Theorem: The solution $u$ in Fourier satisfies $u_j (t) = e^{-i t |j|^2} v_j (\epsilon t) + O(\epsilon)$ for $t \leq \frac{T}{\epsilon}$ where $v_j (\tau)$ solves the resonant system RFNLS.

Proof: integration by parts, contained in other works. The work is done in the Wiener algebra.

Quadruplets…rectangles. In dimension 1, there are no rectangles. therefore, the resonant Hamiltonian only depends upon the actions.

We have preservation of the actions in 1d over a time $\epsilon^{-1}$. If you try to reproduce this numerically, it is quite difficult due to the aliasing issue. Prime numbers help to avoid the aliasing issue…interesting.

Simulating energy Cascades. Consider data supported on 5 modes so that it forms a cross. He’s revisiting the construction I displayed in Napoli! Very cool. Dynamics of the extremal modes can be tracked.

Theorem (Carles-Faou 2010): Let $u_0 (x,y) = 1 + 2 \cos x + 2 \co y. Then….ack slide changed.

After n iterations, the mode that is turned on is such that $|j|^2 = 2^n$. For that mode, we have a lower bound on the size of that Fourier coefficient. I ran a long time simulation the other night. The same initial data (the cross) and he tracked the $H^4$ norm. You start at order 0.1 and after time 2000 you reach order 1.

More numerics needed (with R. Belaouar, CMAP) we are trying to do some very long simulations. This is a different mechanism from CKSTT.

He showed a movie which showed a slowly growing island of activity near the origin. Very cool 5 frequency model starting on a “cross”. The example reminded me of the cartoon version of the CKSTT construction I exposed in Napoli.

Plane Wave Stability (with Lubich and Gauckler)

preprint

When the $L^2$ norm lies inside some typical set. quantified preservation of the super actions….too fast for me to type. Orbital stability of the plane wave.

Plane waves stability:

Phase invariance
$|u_0|^2 is controlled by the $L^2$ norm of the $u_j, j \neq 0$.
Change of variables $(u_0, u_j) \longmapsto (a, \theta, v_j)$ …slide change.

Gauge invariance squeezes out the dependence on $\theta$. Preservation of the $L^2$ norm. The equations close under the change of variables. Even though the change of variables is not symplectic, there is some magic. The system for the $v_j$ turns out to be Hamiltonian. The normal form toolbox is applied and the genericity condition (in measure) is exploited….lots to understand here.

Discussion following the talk among Bambusi, Hani, Faou and me: Why doesn’t this contradict the theorem of Hani? Answer. The plane wave stability time here is limited. Hani’s effect takes place much much later. The FGL result is analogous to Nekoroshev and Hani’s shows diffusion after the stability time. There was a suggestion that there might be a KAM theorem lurking here which would allow the FGL stability type result to persist to infinite time provided that the condition on $\rho$ is implemented more cleverly. Very interesting….

Friday 2011-09-09

Dario Bambusi: Solitons in a numerical algorithm for NLS

(joint work with Erwan Faou and Benoit Grébert)

The point: when you compute you calculate the dynamics of a numerical approximate model of the problem. Solitons can sometimes be destroyed by the algorithm.

Consider focusing NLS on R. There are ground state, particular solutions. We know that these solutions are orbitally stable.

What happens if you try to put on the computer the dynamics of the NLS. We consider a large window and make a space discretization. We have reduced the problem into a finite dimensional system of ODEs. Then, you use a splitting method. You compose a flow associated with the vector field associated with the nonlinearity (called $P$) followed by the linear flow (called $A$).

The Euler method is not a symplectic method. We use an exact method for the nonlinearity and Euler for the linear. The soliton does not persist but is eventually destroyed. Under a CFL condition ($\tau/\mu^2$ is not too large), things improve. If this is too large there is a resonance and bad things can happen. He shows an example with CFL of size 19 which shows the growth of high frequencies. These grow higher and higher and after 3000 steps the soliton is completely lost. On the contrary, if you take a small CFL of size 1.9, then the soliton persists for a long number of iterations.

The question we want to address: Can we explain what is going on analytically here?

Space Discretization: Let $\psi (x)$ be substituted by $\psi_j = \psi (\mu j), ~ \mu \ll 1$ discretization. Substitute the Laplacian by the standard discrete version of it. You then find the usual discrete NLS. To put it on a computer, you cut off the number of points. So, instead of sampling at all points $j \in Z$, you wok on a large interval $[-K, K]$ with Dirichlet boundary conditions.

Time Discretization and preliminaries: Let $\phi^t_X$ denote the flow map of the vector field $X$. We use a splitting method. Substitute the flow of DNLS by $\phi^\tau_{A+P}$. WE have the soliton manifold $S$ and we have the discrete version of the $H^1$ space: $ | \psi |{E\mu}^2 = \mu ( \frac{1}{\mu^2} \sum_j |\psi_j - \psi_{j-1}^2 + \sum_j |\psi_j|^2). $

Theorem:

Assume that

$\mu \ll 1$ and $\tau \ll 1$
let $r \geq 4$ be an integer such that $\frac{r \tau}{\mu^2} < \pi$.

initial datum: $\psi$

If $d(\psi, S) \leq C (\mu + \frac{\tau}{\mu^{1/2}} + \frac{1}{\mu^2} e^{-C\mu K})$ then for $|n| \leq C \tau^{2-r}$ one has $d(()\phi^\tau_Q \circ \phi^\tau_P)^n, S) $ …ack slide changed.

Idea of Proof:

Conserved quantities. We know that $\phi_c$ realizes the minimum of the energy subject to a mass constraint. The minimum is unique assuming even in $x$.
space discretization (Bambusi-Penati): equal characterization with nearby functionals. space cutoff: idem.
time discretization (splitting): there exists a modified energy which is quasiconserved for the algorithm. (Benettin-Girogilli, Faou-grébert).

finite elements; discrete solitons;

Hamiltonian interpolation. Problem: does there exist $Z$ Hamiltonian such that $\phi^\tau_A \circ \phi^\tau_P - \phi^1_{\tau Z} = 0$? That is, can we show that the composition of two Hamiltonian flows is another Hamiltonian flow. (This seems obvious to me but I think the issue has to do with making this claim valid in the discrete setting.)

Galina Perelman: Contracting sphere blow up solutions for the 3D cubic NLS

(joint work with J. Holmer and S. Roudenko)

We consider the cubic focusing NLS in 3D.

conservation of mass, momentum and energy.

virial identity.

$\dot{H}^{1/2}$-critical. Scaling $\psi (t,x) \longmapsto \lambda \psi(\lambda^2 t, \lambda x)$

General facts:

global existence and scattering for small $H^{1/2}$ data.
Virial identity implies existence of blowup solutions
scaling lower bound on the blowup rate.
Merle-Raphaël has a more sophisticated blowup rate. The critical norm explodes faster than a power of the $\log (T-t)$.

Blowup scenarios:

*Self-similar blowup: *

Numerical experiments strongly suggest the existence of self-similar blowup solutions. It is expected that this is true however, the profiles are not in the critical space. Therefore, the asymptotic is valid only locally and we need some cutoff that will grow and will account for the explosion of the critical norm.

Merle-Raphaël-Szeftel: Rigrous justification of the self-similar blwup regime for slightly $L^2$ supercritical NLS.

* Circle blowup solutions (Holmer-Roudenko, Zwiers):*

Consider cylindrical coordinates on $R^3$. $Q$ denotes the ground state of the 2D cubic NLS…..dynamic is stable wrt to $H^1$ initial perturbation preserving the cylindrical symmetry. The idea of this construction is inspired by Raphaël who constructed solutions to the quintic NLS on $R^2$. In cylindrical coordinates, the problem resembles the 2D cubic NLS in $(z,r)$ apart from another term from the Laplacian. The extra term should not contribute to the dynamics provided that the explosion takes place away from the origin. This idea is built on the ring blowup solution of Raphaël. Should also mention Raphaël-Szeftel.

* Contracting sphere blowup solutions:*

Fibich-Gavish-Wang: numerical results an some heuristic arguments suggest the existence of radial finite time blowup solutions which explode on a contracting sphere. Assuming these exist, the behavior of the thickness scaling parameter and the contracting radius parameter can be calculated using the conservation laws.

Theorem: There exists a radial solution $\psi$ to cubic NLS on $R^3$ with $\psi \in C((0,t_0]; H^1)$ which explodes as a contracting sphere with radius $q(t)$ and scaling parameter $\lambda(t)$. with $q \thicksim t^{1/3}$ and $\lambda \thicksim t^{-2/3}$. The blowup rate for the $H^1$ norm is $t^{-2/3}$.

Remark: The choice of 3D cubic NLS is for its simplicity. One might expect that the same result holds true for other $L^2$ supercritical and $H^1$ subcritical problems with adjusted behavior for $\lambda$ and $q$. In our calculation, we use the fact that the nonlinearity is $C^\infty$.

Outline of Proof

Step 1. Construction an arbitrarily good approximate solution (up to any order) Step 2. Construct exact solution with small remainder.

….slides moving fast

New parameter is $E = \lambda^{-1}q^{-2}$.

We expect that $q \thicksim t^{1/3}$ and $E \thicksim 1$ so that $\lambda \thicksim t^{-2/3}$.

Build a formal solution….solvability conditions are trivial for even parameter $j$ but for odd $j$ these are nontrivial and allow to determine the coefficients $q_l$.

Build an approximate solution. We cutoff the iteration process used to define the formal solution at some stage. The error of the approximate solution is quantified to be small, like $t^{2N=2}$, after $N$ steps in the formal iteration.

Construction of an exact solution. Proposition: There exists a solution of the cubic NLS with is $t^N$-close to the approximate solution $\psi^{(N)}$. Also $\psi \in C([t_1, t_0], H^1 \cap ^{-1} L^2)$.

Main theorem follows from the proposition. Snapshots and a profile extraction, LWP….

How to prove the proposition? Bootstrap arguments based on energy type estimate.

Almost conservation of a quantity $G$. Coercivity of $G$. Control of missing directions: Conservation laws control two of them. Two others are controlled using the equation.

Q: Can you build concentric rings that collapse? Or are the parameters rigidly linked?

Maybe. There is some flexibility in the construction…. Matryoshka Doll Blowup?